Theorem caovassg 5679

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovassg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))

Assertion

Ref	Expression
caovassg	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovassg

Step	Hyp	Ref	Expression
1		caovassg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
2	1	ralrimivvva 2444	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
3		oveq1 5539	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	oveq1d 5547	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝑦)𝐹𝑧))
5		oveq1 5539	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝑦𝐹𝑧)))
6	4, 5	eqeq12d 2095	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧))))
7		oveq2 5540	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
8	7	oveq1d 5547	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝑧))
9		oveq1 5539	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
10	9	oveq2d 5548	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝑧)))
11	8, 10	eqeq12d 2095	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧))))
12		oveq2 5540	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝐶))
13		oveq2 5540	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
14	13	oveq2d 5548	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐹(𝐵𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝐶)))
15	12, 14	eqeq12d 2095	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))))
16	6, 11, 15	rspc3v 2716	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))))
17	2, 16	mpan9 275	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caovassd  5680  caovass  5681  grprinvlem  5715  grprinvd  5716  grpridd  5717  iseqsplit  9458  iseqcaopr  9462