Theorem caov12d 6186

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 6161	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
5	4	oveq1d 6015	. 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 6164	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
9	6, 3, 2, 7	caovassd 6164	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶)))
10	5, 8, 9	3eqtr3d 2270	1 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  (class class class)co 6000

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  caov4d  6189  caovimo  6198  ltaddnq  7590  ltexnqq  7591  enq0tr  7617  mullocprlem  7753  1idprl  7773  1idpru  7774  cauappcvgprlemdisj  7834  mulcmpblnrlemg  7923  lttrsr  7945  ltsosr  7947  0idsr  7950  1idsr  7951  recexgt0sr  7956  mulgt0sr  7961  axmulass  8056