Theorem grpoass 30024

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 30018	. . . 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 1142	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
5		oveq1 7419	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
6	5	oveq1d 7427	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝑦)𝐺𝑧))
7		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝑦𝐺𝑧)))
8	6, 7	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧))))
9		oveq2 7420	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
10	9	oveq1d 7427	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝑧))
11		oveq1 7419	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7428	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝑧)))
13	10, 12	eqeq12d 2747	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧))))
14		oveq2 7420	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝐶))
15		oveq2 7420	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	15	oveq2d 7428	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝐶)))
17	14, 16	eqeq12d 2747	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
18	8, 13, 17	rspc3v 3627	. 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 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   × cxp 5674  ran crn 5677  ⟶wf 6539  (class class class)co 7412  GrpOpcgr 30010

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7415  df-grpo 30014

This theorem is referenced by:  grpoidinvlem1  30025  grpoidinvlem2  30026  grpoidinvlem4  30028  grporcan  30039  grpoinvid1  30049  grpoinvid2  30050  grpolcan  30051  grpoinvop  30054  grpomuldivass  30062  grponpcan  30064  ablo32  30070  ablo4  30071  vcm  30097  nvass  30143  hhssabloilem  30782  rngoaass  37086