Theorem mulcmpblnrlem 6334

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 4889	. . . . . . . . 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 2295	. . . . . . . . . . 11 |- x e. _V
6		visset 2295	. . . . . . . . . . 11 |- y e. _V
7	5, 6	mulcompr 6277	. . . . . . . . . 10 |- (x .P. y) = (y .P. x)
8		visset 2295	. . . . . . . . . . 11 |- z e. _V
9	6, 8	distrpr 6284	. . . . . . . . . 10 |- (x .P. (y +P. z)) = ((x .P. y) +P. (x .P. z))
10	2, 3, 4, 7, 9	caoprdistrr 5000	. . . . . . . . 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 5000	. . . . . . . . 9 |- ((B +P. C) .P. F) = ((B .P. F) +P. (C .P. F))
14	1, 10, 13	3eqtr3g 1952	. . . . . . . 8 |- ((A +P. D) = (B +P. C) -> ((A .P. F) +P. (D .P. F)) = ((B .P. F) +P. (C .P. F)))
15	14	opreq1d 4897	. . . . . . 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 4890	. . . . . . . . . 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 6284	. . . . . . . . . 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 6284	. . . . . . . . . 10 |- (C .P. (G +P. R)) = ((C .P. G) +P. (C .P. R))
22	16, 18, 21	3eqtr3g 1952	. . . . . . . . 9 |- ((F +P. S) = (G +P. R) -> ((C .P. F) +P. (C .P. S)) = ((C .P. G) +P. (C .P. R)))
23	22	opreq2d 4898	. . . . . . . 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 4907	. . . . . . . . 9 |- (C .P. F) e. _V
25		oprex 4907	. . . . . . . . 9 |- (C .P. S) e. _V
26	24, 25	addasspr 6276	. . . . . . . 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 1940	. . . . . . 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 1948	. . . . . 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 4907	. . . . . . 7 |- (A .P. F) e. _V
30		oprex 4907	. . . . . . 7 |- (D .P. F) e. _V
31	5, 6	addcompr 6275	. . . . . . 7 |- (x +P. y) = (y +P. x)
32	6, 8	addasspr 6276	. . . . . . 7 |- ((x +P. y) +P. z) = (x +P. (y +P. z))
33	29, 30, 25, 31, 32	caopr32 4993	. . . . . 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 4907	. . . . . . 7 |- (B .P. F) e. _V
35		oprex 4907	. . . . . . 7 |- (C .P. G) e. _V
36		oprex 4907	. . . . . . 7 |- (C .P. R) e. _V
37	34, 35, 36, 31, 32	caopr12 4994	. . . . . 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 1952	. . . . 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 4898	. . . 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 4890	. . . . . . . . . . 11 |- ((F +P. S) = (G +P. R) -> (D .P. (F +P. S)) = (D .P. (G +P. R)))
41	4, 17	distrpr 6284	. . . . . . . . . . 11 |- (D .P. (F +P. S)) = ((D .P. F) +P. (D .P. S))
42	19, 20	distrpr 6284	. . . . . . . . . . 11 |- (D .P. (G +P. R)) = ((D .P. G) +P. (D .P. R))
43	40, 41, 42	3eqtr3g 1952	. . . . . . . . . 10 |- ((F +P. S) = (G +P. R) -> ((D .P. F) +P. (D .P. S)) = ((D .P. G) +P. (D .P. R)))
44	43	opreq2d 4898	. . . . . . . . 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 4907	. . . . . . . . . 10 |- (D .P. G) e. _V
46		oprex 4907	. . . . . . . . . 10 |- (D .P. R) e. _V
47	45, 46	addasspr 6276	. . . . . . . . 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 1946	. . . . . . . 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 4889	. . . . . . . . . 10 |- ((A +P. D) = (B +P. C) -> ((A +P. D) .P. G) = ((B +P. C) .P. G))
50	2, 3, 19, 7, 9	caoprdistrr 5000	. . . . . . . . . 10 |- ((A +P. D) .P. G) = ((A .P. G) +P. (D .P. G))
51	11, 12, 19, 7, 9	caoprdistrr 5000	. . . . . . . . . 10 |- ((B +P. C) .P. G) = ((B .P. G) +P. (C .P. G))
52	49, 50, 51	3eqtr3g 1952	. . . . . . . . 9 |- ((A +P. D) = (B +P. C) -> ((A .P. G) +P. (D .P. G)) = ((B .P. G) +P. (C .P. G)))
53	52	opreq1d 4897	. . . . . . . 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 1951	. . . . . . 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 4907	. . . . . . . 8 |- (A .P. G) e. _V
56		oprex 4907	. . . . . . . 8 |- (D .P. S) e. _V
57	55, 30, 56, 31, 32	caopr12 4994	. . . . . . 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 4907	. . . . . . . 8 |- (B .P. G) e. _V
59	58, 35, 46, 31, 32	caopr32 4993	. . . . . . 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 1952	. . . . . 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 4897	. . . . 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 4907	. . . . . 6 |- ((B .P. F) +P. (C .P. R)) e. _V
63	35, 62	addasspr 6276	. . . . 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 1944	. . . 4 |- (((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)))))
65	39, 64	eqtr4d 1928	. . 3 |- (((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))) = (((D .P. F) +P. ((A .P. G) +P. (D .P. S))) +P. ((B .P. F) +P. (C .P. R))))
66		oprex 4907	. . . 4 |- ((B .P. G) +P. (D .P. R)) e. _V
67		oprex 4907	. . . 4 |- ((A .P. F) +P. (C .P. S)) e. _V
68	66, 67, 30, 31, 32	caopr13 4996	. . 3 |- (((B .P. G) +P. (D .P. R)) +P. (((A .P. F) +P. (C .P. S)) +P. (D .P. F))) = ((D .P. F) +P. (((A .P. F) +P. (C .P. S)) +P. ((B .P. G) +P. (D .P. R))))
69		oprex 4907	. . . 4 |- ((A .P. G) +P. (D .P. S)) e. _V
70	69, 62	addasspr 6276	. . 3 |- (((D .P. F) +P. ((A .P. G) +P. (D .P. S))) +P. ((B .P. F) +P. (C .P. R))) = ((D .P. F) +P. (((A .P. G) +P. (D .P. S)) +P. ((B .P. F) +P. (C .P. R))))
71	65, 68, 70	3eqtr3g 1952	. 2 |- (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> ((D .P. F) +P. (((A .P. F) +P. (C .P. S)) +P. ((B .P. G) +P. (D .P. R)))) = ((D .P. F) +P. (((A .P. G) +P. (D .P. S)) +P. ((B .P. F) +P. (C .P. R)))))
72	29, 25, 58, 31, 32, 46	caopr4 4997	. . 3 |- (((A .P. F) +P. (C .P. S)) +P. ((B .P. G) +P. (D .P. R))) = (((A .P. F) +P. (B .P. G)) +P. ((C .P. S) +P. (D .P. R)))
73	72	opreq2i 4893	. 2 |- ((D .P. F) +P. (((A .P. F) +P. (C .P. S)) +P. ((B .P. G) +P. (D .P. R)))) = ((D .P. F) +P. (((A .P. F) +P. (B .P. G)) +P. ((C .P. S) +P. (D .P. R))))
74	55, 56, 34, 31, 32, 36	caopr42 4999	. . 3 |- (((A .P. G) +P. (D .P. S)) +P. ((B .P. F) +P. (C .P. R))) = (((A .P. G) +P. (B .P. F)) +P. ((C .P. R) +P. (D .P. S)))
75	74	opreq2i 4893	. 2 |- ((D .P. F) +P. (((A .P. G) +P. (D .P. S)) +P. ((B .P. F) +P. (C .P. R)))) = ((D .P. F) +P. (((A .P. G) +P. (B .P. F)) +P. ((C .P. R) +P. (D .P. S))))
76	71, 73, 75	3eqtr3g 1952	1 |- (((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)))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (class class class)co 4884   +P. cpp 6139   .P. cmp 6140

This theorem is referenced by:  mulcmpblnr 6335

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-plp 6240  df-mp 6241

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