Theorem caovdilemd 6033

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 5995	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆)
6		caovdilemd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
7		caovdilemd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
8	2, 6, 7	caovcld 5995	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆)
9		caovdilemd.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
10	1, 5, 8, 9	caovdird 6020	. 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 6001	. . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)))
13	11, 6, 7, 9	caovassd 6001	. . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)))
14	12, 13	oveq12d 5860	. 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
15	10, 14	eqtrd 2198	1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  caovlem2d  6034  addassnqg  7323  addassnq0  7403  axmulass  7814