Theorem caov4d 6274

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
caov4d	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))

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 caov4d

Step	Hyp	Ref	Expression
1		caovd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
2		caovd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
3		caovd.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
4		caovd.com	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5		caovd.ass	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
6	1, 2, 3, 4, 5	caov12d 6271	. . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)))
7	6	oveq2d 6101	. 2 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
8		caovd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
9		caovd.cl	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
10	9, 2, 3	caovcld 6243	. . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆)
11	5, 8, 1, 10	caovassd 6249	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))))
12	9, 1, 3	caovcld 6243	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆)
13	5, 8, 2, 12	caovassd 6249	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
14	7, 11, 13	3eqtr4d 2281	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  caov411d  6275  caov42d  6276  ecopovtrn  6906  ecopovtrng  6909  addcmpblnq  7734  mulcmpblnq  7735  ordpipqqs  7741  distrnqg  7754  ltsonq  7765  ltanqg  7767  ltmnqg  7768  addcmpblnq0  7810  mulcmpblnq0  7811  distrnq0  7826  prarloclemlo  7861  addlocprlemeqgt  7899  addcanprleml  7981  recexprlem1ssl  8000  recexprlem1ssu  8001  mulcmpblnrlemg  8107  distrsrg  8126  ltasrg  8137  mulgt0sr  8145  prsradd  8153  axdistr  8241