Theorem grpoass 30523

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 30517	. . . 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 267	. . 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 1143	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
5		oveq1 7439	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
6	5	oveq1d 7447	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝑦)𝐺𝑧))
7		oveq1 7439	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝑦𝐺𝑧)))
8	6, 7	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧))))
9		oveq2 7440	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
10	9	oveq1d 7447	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝑧))
11		oveq1 7439	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7448	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝑧)))
13	10, 12	eqeq12d 2752	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧))))
14		oveq2 7440	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝐶))
15		oveq2 7440	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	15	oveq2d 7448	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝐶)))
17	14, 16	eqeq12d 2752	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
18	8, 13, 17	rspc3v 3637	. 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 506	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069   × cxp 5682  ran crn 5685  ⟶wf 6556  (class class class)co 7432  GrpOpcgr 30509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-ov 7435  df-grpo 30513

This theorem is referenced by:  grpoidinvlem1  30524  grpoidinvlem2  30525  grpoidinvlem4  30527  grporcan  30538  grpoinvid1  30548  grpoinvid2  30549  grpolcan  30550  grpoinvop  30553  grpomuldivass  30561  grponpcan  30563  ablo32  30569  ablo4  30570  vcm  30596  nvass  30642  hhssabloilem  31281  rngoaass  37922