Theorem caovlem2d 6034

Description: Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)

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)
caovdl2.6	⊢ (𝜑 → 𝑅 ∈ 𝑆)
caovdl2.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovdl2.ass	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovdl2.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)

Assertion

Ref	Expression
caovlem2d	⊢ (𝜑 → ((((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   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovlem2d

Dummy variables 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caovdilemd.cl	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
2		caovdilemd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovdilemd.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
4		caovdilemd.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
5	1, 3, 4	caovcld 5995	. . . 4 ⊢ (𝜑 → (𝐶𝐺𝐻) ∈ 𝑆)
6	1, 2, 5	caovcld 5995	. . 3 ⊢ (𝜑 → (𝐴𝐺(𝐶𝐺𝐻)) ∈ 𝑆)
7		caovdilemd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
8		caovdilemd.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
9	1, 8, 4	caovcld 5995	. . . 4 ⊢ (𝜑 → (𝐷𝐺𝐻) ∈ 𝑆)
10	1, 7, 9	caovcld 5995	. . 3 ⊢ (𝜑 → (𝐵𝐺(𝐷𝐺𝐻)) ∈ 𝑆)
11		caovdl2.6	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
12	1, 8, 11	caovcld 5995	. . . 4 ⊢ (𝜑 → (𝐷𝐺𝑅) ∈ 𝑆)
13	1, 2, 12	caovcld 5995	. . 3 ⊢ (𝜑 → (𝐴𝐺(𝐷𝐺𝑅)) ∈ 𝑆)
14		caovdl2.com	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
15		caovdl2.ass	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
16	1, 3, 11	caovcld 5995	. . . 4 ⊢ (𝜑 → (𝐶𝐺𝑅) ∈ 𝑆)
17	1, 7, 16	caovcld 5995	. . 3 ⊢ (𝜑 → (𝐵𝐺(𝐶𝐺𝑅)) ∈ 𝑆)
18		caovdl2.cl	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
19	6, 10, 13, 14, 15, 17, 18	caov42d 6028	. 2 ⊢ (𝜑 → (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻)))))
20		caovdilemd.com	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
21		caovdilemd.distr	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
22		caovdilemd.ass	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
23	20, 21, 22, 1, 2, 7, 3, 8, 4	caovdilemd 6033	. . 3 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
24	20, 21, 22, 1, 2, 7, 8, 3, 11	caovdilemd 6033	. . 3 ⊢ (𝜑 → (((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅) = ((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅))))
25	23, 24	oveq12d 5860	. 2 ⊢ (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))))
26		simpr1 993	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
27	18	caovclg 5994	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐹𝑠) ∈ 𝑆)
28	27	caovclg 5994	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦𝐹𝑧) ∈ 𝑆)
29	28	3adantr1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦𝐹𝑧) ∈ 𝑆)
30	26, 29	jca 304	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥 ∈ 𝑆 ∧ (𝑦𝐹𝑧) ∈ 𝑆))
31	20	caovcomg 5997	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐺𝑠) = (𝑠𝐺𝑟))
32	31	caovcomg 5997	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ (𝑦𝐹𝑧) ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑦𝐹𝑧)𝐺𝑥))
33	30, 32	syldan 280	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑦𝐹𝑧)𝐺𝑥))
34		3anrot 973	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
35	21	caovdirg 6019	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑟𝐹𝑠)𝐺𝑡) = ((𝑟𝐺𝑡)𝐹(𝑠𝐺𝑡)))
36	35	caovdirg 6019	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑦𝐹𝑧)𝐺𝑥) = ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)))
37	34, 36	sylan2b 285	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦𝐹𝑧)𝐺𝑥) = ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)))
38	20	eqcomd 2171	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
39	38	3adantr3 1148	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
40	31	caovcomg 5997	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
41	40	ancom2s 556	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
42	41	3adantr2 1147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
43	39, 42	oveq12d 5860	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
44	33, 37, 43	3eqtrd 2202	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
45	44, 2, 5, 12	caovdid 6017	. . 3 ⊢ (𝜑 → (𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅))))
46	44, 7, 16, 9	caovdid 6017	. . 3 ⊢ (𝜑 → (𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
47	45, 46	oveq12d 5860	. 2 ⊢ (𝜑 → ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻)))))
48	19, 25, 47	3eqtr4d 2208	1 ⊢ (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(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:  mulasssrg  7699