Theorem caov4d 6049

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 6046	. . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)))
7	6	oveq2d 5881	. 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 6018	. . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆)
11	5, 8, 1, 10	caovassd 6024	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))))
12	9, 1, 3	caovcld 6018	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆)
13	5, 8, 2, 12	caovassd 6024	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
14	7, 11, 13	3eqtr4d 2218	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2146  (class class class)co 5865

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  caov411d  6050  caov42d  6051  ecopovtrn  6622  ecopovtrng  6625  addcmpblnq  7341  mulcmpblnq  7342  ordpipqqs  7348  distrnqg  7361  ltsonq  7372  ltanqg  7374  ltmnqg  7375  addcmpblnq0  7417  mulcmpblnq0  7418  distrnq0  7433  prarloclemlo  7468  addlocprlemeqgt  7506  addcanprleml  7588  recexprlem1ssl  7607  recexprlem1ssu  7608  mulcmpblnrlemg  7714  distrsrg  7733  ltasrg  7744  mulgt0sr  7752  prsradd  7760  axdistr  7848