Theorem caov4d 5963

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 5960	. . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)))
7	6	oveq2d 5798	. 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 5932	. . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆)
11	5, 8, 1, 10	caovassd 5938	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))))
12	9, 1, 3	caovcld 5932	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆)
13	5, 8, 2, 12	caovassd 5938	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
14	7, 11, 13	3eqtr4d 2183	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  caov411d  5964  caov42d  5965  ecopovtrn  6534  ecopovtrng  6537  addcmpblnq  7199  mulcmpblnq  7200  ordpipqqs  7206  distrnqg  7219  ltsonq  7230  ltanqg  7232  ltmnqg  7233  addcmpblnq0  7275  mulcmpblnq0  7276  distrnq0  7291  prarloclemlo  7326  addlocprlemeqgt  7364  addcanprleml  7446  recexprlem1ssl  7465  recexprlem1ssu  7466  mulcmpblnrlemg  7572  distrsrg  7591  ltasrg  7602  mulgt0sr  7610  prsradd  7618  axdistr  7706