Theorem mulcmpblnrlem 3976

Description: Lemma used in lemma showing compatibility of multiplication.

Hypotheses

Ref	Expression
cmpblnr.1	|- A e. V
cmpblnr.2	|- B e. V
cmpblnr.3	|- C e. V
cmpblnr.4	|- D e. V
cmpblnr.5	|- F e. V
cmpblnr.6	|- G e. V
cmpblnr.7	|- R e. V
cmpblnr.8	|- S e. V

Assertion

Ref	Expression
mulcmpblnrlem	|- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> ((D .P F) +P. (((A .P F) +P. (B .P G)) +P. ((C .P S) +P. (D .P R)))) = ((D .P F) +P. (((A .P G) +P. (B .P F)) +P. ((C .P R) +P. (D .P S)))))

Proof of Theorem mulcmpblnrlem

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . . . . . 9 |- ((A +P. D) = (B +P. C) -> ((A +P. D) .P F) = ((B +P. C) .P F))
2		cmpblnr.1	. . . . . . . . . 10 |- A e. V
3		cmpblnr.4	. . . . . . . . . 10 |- D e. V
4		cmpblnr.5	. . . . . . . . . 10 |- F e. V
5		visset 1350	. . . . . . . . . . 11 |- x e. V
6		visset 1350	. . . . . . . . . . 11 |- y e. V
7	5, 6	mulcompr 3919	. . . . . . . . . 10 |- (x .P y) = (y .P x)
8		visset 1350	. . . . . . . . . . 11 |- z e. V
9	6, 8	distrpr 3926	. . . . . . . . . 10 |- (x .P (y +P. z)) = ((x .P y) +P. (x .P z))
10	2, 3, 4, 7, 9	caoprdistrr 3081	. . . . . . . . 9 |- ((A +P. D) .P F) = ((A .P F) +P. (D .P F))
11		cmpblnr.2	. . . . . . . . . 10 |- B e. V
12		cmpblnr.3	. . . . . . . . . 10 |- C e. V
13	11, 12, 4, 7, 9	caoprdistrr 3081	. . . . . . . . 9 |- ((B +P. C) .P F) = ((B .P F) +P. (C .P F))
14	1, 10, 13	3eqtr3g 1146	. . . . . . . 8 |- ((A +P. D) = (B +P. C) -> ((A .P F) +P. (D .P F)) = ((B .P F) +P. (C .P F)))
15	14	opreq1d 3012	. . . . . . 7 |- ((A +P. D) = (B +P. C) -> (((A .P F) +P. (D .P F)) +P. (C .P S)) = (((B .P F) +P. (C .P F)) +P. (C .P S)))
16		opreq2 3007	. . . . . . . . . 10 |- ((F +P. S) = (G +P. R) -> (C .P (F +P. S)) = (C .P (G +P. R)))
17		cmpblnr.8	. . . . . . . . . . 11 |- S e. V
18	4, 17	distrpr 3926	. . . . . . . . . 10 |- (C .P (F +P. S)) = ((C .P F) +P. (C .P S))
19		cmpblnr.6	. . . . . . . . . . 11 |- G e. V
20		cmpblnr.7	. . . . . . . . . . 11 |- R e. V
21	19, 20	distrpr 3926	. . . . . . . . . 10 |- (C .P (G +P. R)) = ((C .P G) +P. (C .P R))
22	16, 18, 21	3eqtr3g 1146	. . . . . . . . 9 |- ((F +P. S) = (G +P. R) -> ((C .P F) +P. (C .P S)) = ((C .P G) +P. (C .P R)))
23	22	opreq2d 3013	. . . . . . . 8 |- ((F +P. S) = (G +P. R) -> ((B .P F) +P. ((C .P F) +P. (C .P S))) = ((B .P F) +P. ((C .P G) +P. (C .P R))))
24		oprex 3018	. . . . . . . . 9 |- (C .P F) e. V
25		oprex 3018	. . . . . . . . 9 |- (C .P S) e. V
26	24, 25	addasspr 3918	. . . . . . . 8 |- (((B .P F) +P. (C .P F)) +P. (C .P S)) = ((B .P F) +P. ((C .P F) +P. (C .P S)))
27	23, 26	syl5eq 1136	. . . . . . 7 |- ((F +P. S) = (G +P. R) -> (((B .P F) +P. (C .P F)) +P. (C .P S)) = ((B .P F) +P. ((C .P G) +P. (C .P R))))
28	15, 27	sylan9eq 1144	. . . . . 6 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> (((A .P F) +P. (D .P F)) +P. (C .P S)) = ((B .P F) +P. ((C .P G) +P. (C .P R))))
29		oprex 3018	. . . . . . 7 |- (A .P F) e. V
30		oprex 3018	. . . . . . 7 |- (D .P F) e. V
31	5, 6	addcompr 3917	. . . . . . 7 |- (x +P. y) = (y +P. x)
32	6, 8	addasspr 3918	. . . . . . 7 |- ((x +P. y) +P. z) = (x +P. (y +P. z))
33	29, 30, 25, 31, 32	caopr32 3074	. . . . . 6 |- (((A .P F) +P. (D .P F)) +P. (C .P S)) = (((A .P F) +P. (C .P S)) +P. (D .P F))
34		oprex 3018	. . . . . . 7 |- (B .P F) e. V
35		oprex 3018	. . . . . . 7 |- (C .P G) e. V
36		oprex 3018	. . . . . . 7 |- (C .P R) e. V
37	34, 35, 36, 31, 32	caopr12 3075	. . . . . 6 |- ((B .P F) +P. ((C .P G) +P. (C .P R))) = ((C .P G) +P. ((B .P F) +P. (C .P R)))
38	28, 33, 37	3eqtr3g 1146	. . . . 5 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> (((A .P F) +P. (C .P S)) +P. (D .P F)) = ((C .P G) +P. ((B .P F) +P. (C .P R))))
39	38	opreq2d 3013	. . . 4 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> (((B .P G) +P. (D .P R)) +P. (((A .P F) +P. (C .P S)) +P. (D .P F))) = (((B .P G) +P. (D .P R)) +P. ((C .P G) +P. ((B .P F) +P. (C .P R)))))
40		opreq2 3007	. . . . . . . . . . 11 |- ((F +P. S) = (G +P. R) -> (D .P (F +P. S)) = (D .P (G +P. R)))
41	4, 17	distrpr 3926	. . . . . . . . . . 11 |- (D .P (F +P. S)) = ((D .P F) +P. (D .P S))
42	19, 20	distrpr 3926	. . . . . . . . . . 11 |- (D .P (G +P. R)) = ((D .P G) +P. (D .P R))
43	40, 41, 42	3eqtr3g 1146	. . . . . . . . . 10 |- ((F +P. S) = (G +P. R) -> ((D .P F) +P. (D .P S)) = ((D .P G) +P. (D .P R)))
44	43	opreq2d 3013	. . . . . . . . 9 |- ((F +P. S) = (G +P. R) -> ((A .P G) +P. ((D .P F) +P. (D .P S))) = ((A .P G) +P. ((D .P G) +P. (D .P R))))
45		oprex 3018	. . . . . . . . . 10 |- (D .P G) e. V
46		oprex 3018	. . . . . . . . . 10 |- (D .P R) e. V
47	45, 46	addasspr 3918	. . . . . . . . 9 |- (((A .P G) +P. (D .P G)) +P. (D .P R)) = ((A .P G) +P. ((D .P G) +P. (D .P R)))
48	44, 47	syl6eqr 1142	. . . . . . . 8 |- ((F +P. S) = (G +P. R) -> ((A .P G) +P. ((D .P F) +P. (D .P S))) = (((A .P G) +P. (D .P G)) +P. (D .P R)))
49		opreq1 3006	. . . . . . . . . 10 |- ((A +P. D) = (B +P. C) -> ((A +P. D) .P G) = ((B +P. C) .P G))
50	2, 3, 19, 7, 9	caoprdistrr 3081	. . . . . . . . . 10 |- ((A +P. D) .P G) = ((A .P G) +P. (D .P G))
51	11, 12, 19, 7, 9	caoprdistrr 3081	. . . . . . . . . 10 |- ((B +P. C) .P G) = ((B .P G) +P. (C .P G))
52	49, 50, 51	3eqtr3g 1146	. . . . . . . . 9 |- ((A +P. D) = (B +P. C) -> ((A .P G) +P. (D .P G)) = ((B .P G) +P. (C .P G)))
53	52	opreq1d 3012	. . . . . . . 8 |- ((A +P. D) = (B +P. C) -> (((A .P G) +P. (D .P G)) +P. (D .P R)) = (((B .P G) +P. (C .P G)) +P. (D .P R)))
54	48, 53	sylan9eqr 1145	. . . . . . 7 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> ((A .P G) +P. ((D .P F) +P. (D .P S))) = (((B .P G) +P. (C .P G)) +P. (D .P R)))
55		oprex 3018	. . . . . . . 8 |- (A .P G) e. V
56		oprex 3018	. . . . . . . 8 |- (D .P S) e. V
57	55, 30, 56, 31, 32	caopr12 3075	. . . . . . 7 |- ((A .P G) +P. ((D .P F) +P. (D .P S))) = ((D .P F) +P. ((A .P G) +P. (D .P S)))
58		oprex 3018	. . . . . . . 8 |- (B .P G) e. V
59	58, 35, 46, 31, 32	caopr32 3074	. . . . . . 7 |- (((B .P G) +P. (C .P G)) +P. (D .P R)) = (((B .P G) +P. (D .P R)) +P. (C .P G))
60	54, 57, 59	3eqtr3g 1146	. . . . . 6 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> ((D .P F) +P. ((A .P G) +P. (D .P S))) = (((B .P G) +P. (D .P R)) +P. (C .P G)))
61	60	opreq1d 3012	. . . . 5 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> (((D .P F) +P. ((A .P G) +P. (D .P S))) +P. ((B .P F) +P. (C .P R))) = ((((B .P G) +P. (D .P R)) +P. (C .P G)) +P. ((B .P F) +P. (C .P R))))
62		oprex 3018	. . . . . 6 |- ((B .P F) +P. (C .P R)) e. V
63	35, 62	addasspr 3918	. . . . 5 |- ((((B .P G) +P. (D .P R)) +P. (C .P G)) +P. ((B .P F) +P. (C .P R))) = (((B .P G) +P. (D .P R)) +P. ((C .P G) +P. ((B .P F) +P. (C .P R))))
64	61, 63	syl6eq 1140	. . . 4 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) ->