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Theorem caovlem2 7373
Description: Lemma used in 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))
caovdl2.6 𝑅 ∈ V
caovdl2.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caovdl2.ass ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
Assertion
Ref Expression
caovlem2 ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(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   𝑥,𝑅,𝑦,𝑧

Proof of Theorem caovlem2
StepHypRef Expression
1 ovex 7178 . . 3 (𝐴𝐺(𝐶𝐺𝐻)) ∈ V
2 ovex 7178 . . 3 (𝐵𝐺(𝐷𝐺𝐻)) ∈ V
3 ovex 7178 . . 3 (𝐴𝐺(𝐷𝐺𝑅)) ∈ V
4 caovdl2.com . . 3 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5 caovdl2.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
6 ovex 7178 . . 3 (𝐵𝐺(𝐶𝐺𝑅)) ∈ V
71, 2, 3, 4, 5, 6caov42 7370 . 2 (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
8 caovdir.1 . . . 4 𝐴 ∈ V
9 caovdir.2 . . . 4 𝐵 ∈ V
10 caovdir.3 . . . 4 𝐶 ∈ V
11 caovdir.com . . . 4 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
12 caovdir.distr . . . 4 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
13 caovdl.4 . . . 4 𝐷 ∈ V
14 caovdl.5 . . . 4 𝐻 ∈ V
15 caovdl.ass . . . 4 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))
168, 9, 10, 11, 12, 13, 14, 15caovdilem 7372 . . 3 (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
17 caovdl2.6 . . . 4 𝑅 ∈ V
188, 9, 13, 11, 12, 10, 17, 15caovdilem 7372 . . 3 (((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅) = ((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))
1916, 18oveq12i 7157 . 2 ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅))))
20 ovex 7178 . . . 4 (𝐶𝐺𝐻) ∈ V
21 ovex 7178 . . . 4 (𝐷𝐺𝑅) ∈ V
228, 20, 21, 12caovdi 7356 . . 3 (𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))
23 ovex 7178 . . . 4 (𝐶𝐺𝑅) ∈ V
24 ovex 7178 . . . 4 (𝐷𝐺𝐻) ∈ V
259, 23, 24, 12caovdi 7356 . . 3 (𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
2622, 25oveq12i 7157 . 2 ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
277, 19, 263eqtr4i 2851 1 ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  mulasssr  10500
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