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      "title": "Lagrange's Theorem at a glance",
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      "transcript": "GROUP THEORY · SUBGROUP CARDINALITY\nLAGRANGE'S THEOREM\ns ≤ α\nNat.card s ∣ Nat.card α\nSUBGROUP\nEQUAL COSETS\nAMBIENT GROUP\nFINITE-CASE INTUITION\nTranslated cosets have equal size.\nEXACT MATHLIB FORM\nNo explicit finiteness hypothesis.\nNat.card is 0 on infinite types.\nEXACT SCOPE\nDivisibility of Nat.card. Not an index formula or element-order theorem."
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      "description": "A central emerald seven-node cluster is repeated as congruent cobalt-and-ivory clusters around a larger circle. Fine gold arrows link the equal pieces, presenting the familiar coset intuition behind the divisibility statement.",
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      "title": "Equal coset cluster schematic",
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    "walkthrough": {
      "introduction": "Mathlib obtains the exact Nat.card product identity from the equivalence between the ambient group and quotient–subgroup pairs, then simplifies that identity to the divisibility statement. The familiar equal-coset picture is finite-case intuition; the declaration itself has no explicit finiteness hypothesis.",
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          "alt": "Cobalt quotient medallions pair by gold threads with identical seven-node forest-green subgroup rosettes, which assemble inside one oval ambient-group field.",
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          "caption": "The quotient–subgroup product equivalence pairs each coset position with one subgroup element, yielding the ambient natural-cardinality product.",
          "description": "The equivalence Subgroup.groupEquivQuotientProdSubgroup identifies the ambient group with the product of its left-coset quotient and the subgroup. Nat.card_congr together with Nat.card_prod converts that equivalence into the exact natural-cardinality product identity.",
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          "title": "Pair quotient positions with subgroup elements",
          "width": 1536,
          "equation": "Nat.card α = Nat.card (α ⧸ s) * Nat.card s",
          "kind": "proof_walkthrough_page",
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        {
          "alt": "One seven-node green subgroup rosette feeds a gold path through a field assembled from identical whole rosettes, with a faint row continuing near the upper boundary.",
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          "caption": "Because the ambient natural cardinality is a whole multiple of the subgroup natural cardinality, the subgroup cardinality divides it.",
          "description": "The final simp step rewrites the product identity and applies dvd_mul_left, using Nat.card of the quotient as the multiplier. This proves Nat.card s divides Nat.card α without adding a finiteness hypothesis or asserting a separate index formula.",
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          "sha256": "sha256:a5b02d42d864a23dc1ceb0a0043316160659d2ba9b8895623e34efe100096404",
          "title": "Read the product identity as divisibility",
          "width": 1536,
          "equation": "Nat.card s ∣ Nat.card α",
          "kind": "proof_walkthrough_page",
          "sequenceIndex": 2
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      ],
      "title": "From quotient–subgroup pairs to divisibility"
    },
    "whyLandmark": "Lagrange's theorem is one of the first structural results in group theory. In the familiar finite case it tightly constrains possible subgroup sizes and element orders."
  },
  "nonClaims": [
    "Proof Atlas did not originate Lagrange's theorem or Mathlib's declaration.",
    "The local run reproduces the exact upstream declaration; it is not a local wrapper.",
    "A successful local check does not by itself create an accepted Atlas result."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.lagrange-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "The natural-number cardinality of a subgroup divides that of its ambient group.",
  "targetId": "target.library.mathlib.lagrange-theorem.v001",
  "title": "Lagrange's Theorem",
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