Theorem caovdilem 7633

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

Step	Hyp	Ref	Expression
1		ovex 7431	. . 3 ⊢ (𝐴𝐺𝐶) ∈ V
2		ovex 7431	. . 3 ⊢ (𝐵𝐺𝐷) ∈ V
3		caovdl.5	. . 3 ⊢ 𝐻 ∈ V
4		caovdir.com	. . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
5		caovdir.distr	. . 3 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
6	1, 2, 3, 4, 5	caovdir 7632	. 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))
10	7, 8, 3, 9	caovass 7598	. . 3 ⊢ ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻))
11		caovdir.2	. . . 4 ⊢ 𝐵 ∈ V
12		caovdl.4	. . . 4 ⊢ 𝐷 ∈ V
13	11, 12, 3, 9	caovass 7598	. . 3 ⊢ ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻))
14	10, 13	oveq12i 7410	. 2 ⊢ (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
15	6, 14	eqtri 2787	1 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))

Syntax hints:   = wceq 1562   ∈ wcel 2144  Vcvv 3456  (class class class)co 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by:  caovlem2  7634  axmulass  11117