Theorem caovlem2d 5635

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 5596	. . . 4 ⊢ (φ → (𝐶𝐺𝐻) ∈ 𝑆)
6	1, 2, 5	caovcld 5596	. . 3 ⊢ (φ → (A𝐺(𝐶𝐺𝐻)) ∈ 𝑆)
7		caovdilemd.b	. . . 4 ⊢ (φ → B ∈ 𝑆)
8		caovdilemd.d	. . . . 5 ⊢ (φ → 𝐷 ∈ 𝑆)
9	1, 8, 4	caovcld 5596	. . . 4 ⊢ (φ → (𝐷𝐺𝐻) ∈ 𝑆)
10	1, 7, 9	caovcld 5596	. . 3 ⊢ (φ → (B𝐺(𝐷𝐺𝐻)) ∈ 𝑆)
11		caovdl2.6	. . . . 5 ⊢ (φ → 𝑅 ∈ 𝑆)
12	1, 8, 11	caovcld 5596	. . . 4 ⊢ (φ → (𝐷𝐺𝑅) ∈ 𝑆)
13	1, 2, 12	caovcld 5596	. . 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 5596	. . . 4 ⊢ (φ → (𝐶𝐺𝑅) ∈ 𝑆)
17	1, 7, 16	caovcld 5596	. . 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 5629	. 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 5634	. . 3 ⊢ (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))
24	20, 21, 22, 1, 2, 7, 8, 3, 11	caovdilemd 5634	. . 3 ⊢ (φ → (((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅) = ((A𝐺(𝐷𝐺𝑅))𝐹(B𝐺(𝐶𝐺𝑅))))
25	23, 24	oveq12d 5473	. 2 ⊢ (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = (((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻)))𝐹((A𝐺(𝐷𝐺𝑅))𝐹(B𝐺(𝐶𝐺𝑅)))))
26		simpr1 909	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → x ∈ 𝑆)
27	18	caovclg 5595	. . . . . . . . 9 ⊢ ((φ ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐹𝑠) ∈ 𝑆)
28	27	caovclg 5595	. . . . . . . 8 ⊢ ((φ ∧ (y ∈ 𝑆 ∧ z ∈ 𝑆)) → (y𝐹z) ∈ 𝑆)
29	28	3adantr1 1062	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (y𝐹z) ∈ 𝑆)
30	26, 29	jca 290	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (x ∈ 𝑆 ∧ (y𝐹z) ∈ 𝑆))
31	20	caovcomg 5598	. . . . . . 7 ⊢ ((φ ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (𝑟𝐺𝑠) = (𝑠𝐺𝑟))
32	31	caovcomg 5598	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ (y𝐹z) ∈ 𝑆)) → (x𝐺(y𝐹z)) = ((y𝐹z)𝐺x))
33	30, 32	syldan 266	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (x𝐺(y𝐹z)) = ((y𝐹z)𝐺x))
34		3anrot 889	. . . . . 6 ⊢ ((x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆) ↔ (y ∈ 𝑆 ∧ z ∈ 𝑆 ∧ x ∈ 𝑆))
35	21	caovdirg 5620	. . . . . . 7 ⊢ ((φ ∧ (𝑟 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑟𝐹𝑠)𝐺𝑡) = ((𝑟𝐺𝑡)𝐹(𝑠𝐺𝑡)))
36	35	caovdirg 5620	. . . . . 6 ⊢ ((φ ∧ (y ∈ 𝑆 ∧ z ∈ 𝑆 ∧ x ∈ 𝑆)) → ((y𝐹z)𝐺x) = ((y𝐺x)𝐹(z𝐺x)))
37	34, 36	sylan2b 271	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((y𝐹z)𝐺x) = ((y𝐺x)𝐹(z𝐺x)))
38	20	eqcomd 2042	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (y𝐺x) = (x𝐺y))
39	38	3adantr3 1064	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (y𝐺x) = (x𝐺y))
40	31	caovcomg 5598	. . . . . . . 8 ⊢ ((φ ∧ (z ∈ 𝑆 ∧ x ∈ 𝑆)) → (z𝐺x) = (x𝐺z))
41	40	ancom2s 500	. . . . . . 7 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ z ∈ 𝑆)) → (z𝐺x) = (x𝐺z))
42	41	3adantr2 1063	. . . . . 6 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (z𝐺x) = (x𝐺z))
43	39, 42	oveq12d 5473	. . . . 5 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((y𝐺x)𝐹(z𝐺x)) = ((x𝐺y)𝐹(x𝐺z)))
44	33, 37, 43	3eqtrd 2073	. . . 4 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z)))
45	44, 2, 5, 12	caovdid 5618	. . 3 ⊢ (φ → (A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((A𝐺(𝐶𝐺𝐻))𝐹(A𝐺(𝐷𝐺𝑅))))
46	44, 7, 16, 9	caovdid 5618	. . 3 ⊢ (φ → (B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((B𝐺(𝐶𝐺𝑅))𝐹(B𝐺(𝐷𝐺𝐻))))
47	45, 46	oveq12d 5473	. 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 2079	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 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  (class class class)co 5455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458

This theorem is referenced by:  mulasssrg  6686