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Theorem grpoass 28207
Description: A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoass ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Proof of Theorem grpoass
Dummy variables 𝑥 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . . . 5 𝑋 = ran 𝐺
21isgrpo 28201 . . . 4 (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
32ibi 268 . . 3 (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
43simp2d 1135 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
5 oveq1 7152 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
65oveq1d 7160 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝑦)𝐺𝑧))
7 oveq1 7152 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝑦𝐺𝑧)))
86, 7eqeq12d 2834 . . 3 (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧))))
9 oveq2 7153 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
109oveq1d 7160 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝑧))
11 oveq1 7152 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
1211oveq2d 7161 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝑧)))
1310, 12eqeq12d 2834 . . 3 (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧))))
14 oveq2 7153 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝐶))
15 oveq2 7153 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
1615oveq2d 7161 . . . 4 (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝐶)))
1714, 16eqeq12d 2834 . . 3 (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
188, 13, 17rspc3v 3633 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
194, 18mpan9 507 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136   × cxp 5546  ran crn 5549  wf 6344  (class class class)co 7145  GrpOpcgr 28193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-grpo 28197
This theorem is referenced by:  grpoidinvlem1  28208  grpoidinvlem2  28209  grpoidinvlem4  28211  grporcan  28222  grpoinvid1  28232  grpoinvid2  28233  grpolcan  28234  grpoinvop  28237  grpomuldivass  28245  grponpcan  28247  ablo32  28253  ablo4  28254  vcm  28280  nvass  28326  hhssabloilem  28965  rngoaass  35073
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