Theorem caov42d 6191

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)))
caovd.4	⊢ (𝜑 → 𝐷 ∈ 𝑆)
caovd.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)

Assertion

Ref	Expression
caov42d	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)))

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

Proof of Theorem caov42d

Step	Hyp	Ref	Expression
1		caovd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
2		caovd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
3		caovd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
4		caovd.com	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5		caovd.ass	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
6		caovd.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
7		caovd.cl	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8	1, 2, 3, 4, 5, 6, 7	caov4d 6189	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))
9	4, 2, 6	caovcomd 6161	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) = (𝐷𝐹𝐵))
10	9	oveq2d 6016	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)))
11	8, 10	eqtrd 2262	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)))

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:  caovlem2d  6197  mulcmpblnrlemg  7923  ltasrg  7953  axmulass  8056