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Theorem caovdir 7378
 Description: Reverse distributive law. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caovdir.1 𝐴 ∈ V
caovdir.2 𝐵 ∈ V
caovdir.3 𝐶 ∈ V
caovdir.com (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
caovdir.distr (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
Assertion
Ref Expression
caovdir ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem caovdir
StepHypRef Expression
1 caovdir.3 . . 3 𝐶 ∈ V
2 caovdir.1 . . 3 𝐴 ∈ V
3 caovdir.2 . . 3 𝐵 ∈ V
4 caovdir.distr . . 3 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
51, 2, 3, 4caovdi 7363 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6 ovex 7183 . . 3 (𝐴𝐹𝐵) ∈ V
7 caovdir.com . . 3 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
81, 6, 7caovcom 7341 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
91, 2, 7caovcom 7341 . . 3 (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
101, 3, 7caovcom 7341 . . 3 (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
119, 10oveq12i 7162 . 2 ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
125, 8, 113eqtr3i 2789 1 ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  (class class class)co 7150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153 This theorem is referenced by:  caovdilem  7379  adderpqlem  10414  addassnq  10418  prlem934  10493  prlem936  10507  recexsrlem  10563  mulgt0sr  10565
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