Theorem caovassg 7631

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 3203	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
3		oveq1 7438	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	oveq1d 7446	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝑦)𝐹𝑧))
5		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝑦𝐹𝑧)))
6	4, 5	eqeq12d 2751	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧))))
7		oveq2 7439	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
8	7	oveq1d 7446	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝑧))
9		oveq1 7438	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
10	9	oveq2d 7447	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝑧)))
11	8, 10	eqeq12d 2751	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧))))
12		oveq2 7439	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝐶))
13		oveq2 7439	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
14	13	oveq2d 7447	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐹(𝐵𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝐶)))
15	12, 14	eqeq12d 2751	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))))
16	6, 11, 15	rspc3v 3638	. 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 506	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  caovassd  7632  caovass  7633  seqsplit  14073  seqcaopr  14077  seqf1olem2  14080  grpinvalem  18699  grpinva  18700  grprida  18701