Theorem grpoass 28282

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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Proof of Theorem grpoass

Dummy variables 𝑥 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpfo.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2	1	isgrpo 28276	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
3	2	ibi 269	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
4	3	simp2d 1139	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
5		oveq1 7165	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
6	5	oveq1d 7173	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝑦)𝐺𝑧))
7		oveq1 7165	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝑦𝐺𝑧)))
8	6, 7	eqeq12d 2839	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧))))
9		oveq2 7166	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
10	9	oveq1d 7173	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝑧))
11		oveq1 7165	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7174	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝑧)))
13	10, 12	eqeq12d 2839	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧))))
14		oveq2 7166	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝐶))
15		oveq2 7166	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	15	oveq2d 7174	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝐶)))
17	14, 16	eqeq12d 2839	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
18	8, 13, 17	rspc3v 3638	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
19	4, 18	mpan9 509	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   × cxp 5555  ran crn 5558  ⟶wf 6353  (class class class)co 7158  GrpOpcgr 28268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-ov 7161  df-grpo 28272

This theorem is referenced by:  grpoidinvlem1  28283  grpoidinvlem2  28284  grpoidinvlem4  28286  grporcan  28297  grpoinvid1  28307  grpoinvid2  28308  grpolcan  28309  grpoinvop  28312  grpomuldivass  28320  grponpcan  28322  ablo32  28328  ablo4  28329  vcm  28355  nvass  28401  hhssabloilem  29040  rngoaass  35194