Theorem caovdilemd 5771

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 5733	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆)
6		caovdilemd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
7		caovdilemd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
8	2, 6, 7	caovcld 5733	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆)
9		caovdilemd.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
10	1, 5, 8, 9	caovdird 5758	. 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 5739	. . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)))
13	11, 6, 7, 9	caovassd 5739	. . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)))
14	12, 13	oveq12d 5609	. 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
15	10, 14	eqtrd 2115	1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  (class class class)co 5591

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4934  df-fv 4977  df-ov 5594

This theorem is referenced by:  caovlem2d  5772  addassnqg  6844  addassnq0  6924  axmulass  7311