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Theorem caovdilem 6854
 Description: Lemma used by real number construction. (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))
caovdl.4 𝐷 ∈ V
caovdl.5 𝐻 ∈ V
caovdl.ass ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))
Assertion
Ref Expression
caovdilem (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧

Proof of Theorem caovdilem
StepHypRef Expression
1 ovex 6663 . . 3 (𝐴𝐺𝐶) ∈ V
2 ovex 6663 . . 3 (𝐵𝐺𝐷) ∈ V
3 caovdl.5 . . 3 𝐻 ∈ V
4 caovdir.com . . 3 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
5 caovdir.distr . . 3 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
61, 2, 3, 4, 5caovdir 6853 . 2 (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻))
7 caovdir.1 . . . 4 𝐴 ∈ V
8 caovdir.3 . . . 4 𝐶 ∈ V
9 caovdl.ass . . . 4 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))
107, 8, 3, 9caovass 6819 . . 3 ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻))
11 caovdir.2 . . . 4 𝐵 ∈ V
12 caovdl.4 . . . 4 𝐷 ∈ V
1311, 12, 3, 9caovass 6819 . . 3 ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻))
1410, 13oveq12i 6647 . 2 (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
156, 14eqtri 2642 1 (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  Vcvv 3195  (class class class)co 6635 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638 This theorem is referenced by:  caovlem2  6855  axmulass  9963
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