Theorem grpoass 30968

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 30962	. . . 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 270	. . 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 1161	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
5		oveq1 7423	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
6	5	oveq1d 7431	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝑦)𝐺𝑧))
7		oveq1 7423	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝑦𝐺𝑧)))
8	6, 7	eqeq12d 2778	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧))))
9		oveq2 7424	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
10	9	oveq1d 7431	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝑧))
11		oveq1 7423	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7432	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝑧)))
13	10, 12	eqeq12d 2778	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐺𝑧) = (𝐴𝐺(𝑦𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧))))
14		oveq2 7424	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐺𝑧) = ((𝐴𝐺𝐵)𝐺𝐶))
15		oveq2 7424	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	15	oveq2d 7432	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐺𝑧)) = (𝐴𝐺(𝐵𝐺𝐶)))
17	14, 16	eqeq12d 2778	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐺𝑧) = (𝐴𝐺(𝐵𝐺𝑧)) ↔ ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))))
18	8, 13, 17	rspc3v 3595	. 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 516	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   × cxp 5657  ran crn 5660  ⟶wf 6533  (class class class)co 7416  GrpOpcgr 30954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7419  df-grpo 30958

This theorem is used by:  grpoidinvlem1  30969  grpoidinvlem2  30970  grpoidinvlem4  30972  grporcan  30983  grpoinvid1  30993  grpoinvid2  30994  grpolcan  30995  grpoinvop  30998  grpomuldivass  31006  grponpcan  31008  ablo32  31014  ablo4  31015  vcm  31041  nvass  31087  hhssabloilem  31726  rngoaass  38649