Theorem caov12d 6244

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovd.1	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovd.2	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovd.3	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovd.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovd.ass	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))

Assertion

Ref	Expression
caov12d	⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))

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 caov12d

Step	Hyp	Ref	Expression
1		caovd.com	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2		caovd.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4	1, 2, 3	caovcomd 6219	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
5	4	oveq1d 6073	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶))
6		caovd.ass	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7		caovd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
8	6, 2, 3, 7	caovassd 6222	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
9	6, 3, 2, 7	caovassd 6222	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶)))
10	5, 8, 9	3eqtr3d 2275	1 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  (class class class)co 6058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  caov4d  6247  caovimo  6256  ltaddnq  7738  ltexnqq  7739  enq0tr  7765  mullocprlem  7901  1idprl  7921  1idpru  7922  cauappcvgprlemdisj  7982  mulcmpblnrlemg  8071  lttrsr  8093  ltsosr  8095  0idsr  8098  1idsr  8099  recexgt0sr  8104  mulgt0sr  8109  axmulass  8204