Theorem caovdilemd 6196

Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)

Hypotheses

Ref	Expression
caovdilemd.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
caovdilemd.distr	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
caovdilemd.ass	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
caovdilemd.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
caovdilemd.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovdilemd.b	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovdilemd.c	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovdilemd.d	⊢ (𝜑 → 𝐷 ∈ 𝑆)
caovdilemd.h	⊢ (𝜑 → 𝐻 ∈ 𝑆)

Assertion

Ref	Expression
caovdilemd	⊢ (𝜑 → (((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   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdilemd

Step	Hyp	Ref	Expression
1		caovdilemd.distr	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
2		caovdilemd.cl	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
3		caovdilemd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
4		caovdilemd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
5	2, 3, 4	caovcld 6158	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆)
6		caovdilemd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
7		caovdilemd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
8	2, 6, 7	caovcld 6158	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆)
9		caovdilemd.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
10	1, 5, 8, 9	caovdird 6183	. 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)))
11		caovdilemd.ass	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
12	11, 3, 4, 9	caovassd 6164	. . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)))
13	11, 6, 7, 9	caovassd 6164	. . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)))
14	12, 13	oveq12d 6018	. 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
15	10, 14	eqtrd 2262	1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  (class class class)co 6000

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  caovlem2d  6197  addassnqg  7565  addassnq0  7645  axmulass  8056