Theorem caovlem2d 6045

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 6006	. . . 4 ⊢ (𝜑 → (𝐶𝐺𝐻) ∈ 𝑆)
6	1, 2, 5	caovcld 6006	. . 3 ⊢ (𝜑 → (𝐴𝐺(𝐶𝐺𝐻)) ∈ 𝑆)
7		caovdilemd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
8		caovdilemd.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
9	1, 8, 4	caovcld 6006	. . . 4 ⊢ (𝜑 → (𝐷𝐺𝐻) ∈ 𝑆)
10	1, 7, 9	caovcld 6006	. . 3 ⊢ (𝜑 → (𝐵𝐺(𝐷𝐺𝐻)) ∈ 𝑆)
11		caovdl2.6	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
12	1, 8, 11	caovcld 6006	. . . 4 ⊢ (𝜑 → (𝐷𝐺𝑅) ∈ 𝑆)
13	1, 2, 12	caovcld 6006	. . 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 6006	. . . 4 ⊢ (𝜑 → (𝐶𝐺𝑅) ∈ 𝑆)
17	1, 7, 16	caovcld 6006	. . 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 6039	. 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 6044	. . 3 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
24	20, 21, 22, 1, 2, 7, 8, 3, 11	caovdilemd 6044	. . 3 ⊢ (𝜑 → (((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅) = ((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅))))
25	23, 24	oveq12d 5871	. 2 ⊢ (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))))
26		simpr1 998	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
27	18	caovclg 6005	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐹𝑠) ∈ 𝑆)
28	27	caovclg 6005	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦𝐹𝑧) ∈ 𝑆)
29	28	3adantr1 1151	. . . . . . 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 6008	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐺𝑠) = (𝑠𝐺𝑟))
32	31	caovcomg 6008	. . . . . 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 978	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ↔ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
35	21	caovdirg 6030	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑟𝐹𝑠)𝐺𝑡) = ((𝑟𝐺𝑡)𝐹(𝑠𝐺𝑡)))
36	35	caovdirg 6030	. . . . . 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 2176	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
39	38	3adantr3 1153	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
40	31	caovcomg 6008	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
41	40	ancom2s 561	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
42	41	3adantr2 1152	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
43	39, 42	oveq12d 5871	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
44	33, 37, 43	3eqtrd 2207	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
45	44, 2, 5, 12	caovdid 6028	. . 3 ⊢ (𝜑 → (𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅))))
46	44, 7, 16, 9	caovdid 6028	. . 3 ⊢ (𝜑 → (𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
47	45, 46	oveq12d 5871	. 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 2213	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 973   = wceq 1348   ∈ wcel 2141  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  mulasssrg  7720