Theorem caov12d 6109

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 6084	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
5	4	oveq1d 5940	. 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 6087	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
9	6, 3, 2, 7	caovassd 6087	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶)))
10	5, 8, 9	3eqtr3d 2237	1 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  (class class class)co 5925

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  caov4d  6112  caovimo  6121  ltaddnq  7491  ltexnqq  7492  enq0tr  7518  mullocprlem  7654  1idprl  7674  1idpru  7675  cauappcvgprlemdisj  7735  mulcmpblnrlemg  7824  lttrsr  7846  ltsosr  7848  0idsr  7851  1idsr  7852  recexgt0sr  7857  mulgt0sr  7862  axmulass  7957