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Theorem caov4d 6085
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 (𝜑𝐴𝑆)
caovd.2 (𝜑𝐵𝑆)
caovd.3 (𝜑𝐶𝑆)
caovd.com ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovd.ass ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovd.4 (𝜑𝐷𝑆)
caovd.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (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
StepHypRef Expression
1 caovd.2 . . . 4 (𝜑𝐵𝑆)
2 caovd.3 . . . 4 (𝜑𝐶𝑆)
3 caovd.4 . . . 4 (𝜑𝐷𝑆)
4 caovd.com . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5 caovd.ass . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
61, 2, 3, 4, 5caov12d 6082 . . 3 (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)))
76oveq2d 5916 . 2 (𝜑 → (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
8 caovd.1 . . 3 (𝜑𝐴𝑆)
9 caovd.cl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
109, 2, 3caovcld 6054 . . 3 (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆)
115, 8, 1, 10caovassd 6060 . 2 (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))))
129, 1, 3caovcld 6054 . . 3 (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆)
135, 8, 2, 12caovassd 6060 . 2 (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))))
147, 11, 133eqtr4d 2232 1 (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  (class class class)co 5900
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-iota 5199  df-fv 5246  df-ov 5903
This theorem is referenced by:  caov411d  6086  caov42d  6087  ecopovtrn  6662  ecopovtrng  6665  addcmpblnq  7401  mulcmpblnq  7402  ordpipqqs  7408  distrnqg  7421  ltsonq  7432  ltanqg  7434  ltmnqg  7435  addcmpblnq0  7477  mulcmpblnq0  7478  distrnq0  7493  prarloclemlo  7528  addlocprlemeqgt  7566  addcanprleml  7648  recexprlem1ssl  7667  recexprlem1ssu  7668  mulcmpblnrlemg  7774  distrsrg  7793  ltasrg  7804  mulgt0sr  7812  prsradd  7820  axdistr  7908
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