Theorem mulcmpblnrlem 3976

Description: Lemma used in lemma showing compatibility of multiplication.

Hypotheses

Ref	Expression
cmpblnr.1	⊢ A ∈ V
cmpblnr.2	⊢ B ∈ V
cmpblnr.3	⊢ C ∈ V
cmpblnr.4	⊢ D ∈ V
cmpblnr.5	⊢ F ∈ V
cmpblnr.6	⊢ G ∈ V
cmpblnr.7	⊢ R ∈ V
cmpblnr.8	⊢ S ∈ 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 ∈ V
3		cmpblnr.4	. . . . . . . . . 10 ⊢ D ∈ V
4		cmpblnr.5	. . . . . . . . . 10 ⊢ F ∈ V
5		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
6		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
7	5, 6	mulcompr 3919	. . . . . . . . . 10 ⊢ (x ·P y) = (y ·P x)
8		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ 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 ∈ V
12		cmpblnr.3	. . . . . . . . . 10 ⊢ C ∈ 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 ∈ V
18	4, 17	distrpr 3926	. . . . . . . . . 10 ⊢ (C ·P (F +P S)) = ((C ·P F) +P (C ·P S))
19		cmpblnr.6	. . . . . . . . . . 11 ⊢ G ∈ V
20		cmpblnr.7	. . . . . . . . . . 11 ⊢ R ∈ 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) ∈ V
25		oprex 3018	. . . . . . . . 9 ⊢ (C ·P S) ∈ 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) ∈ V
30		oprex 3018	. . . . . . 7 ⊢ (D ·P F) ∈ 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) ∈ V
35		oprex 3018	. . . . . . 7 ⊢ (C ·P G) ∈ V
36		oprex 3018	. . . . . . 7 ⊢ (C ·P R) ∈ 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) ∈ V
46		oprex 3018	. . . . . . . . . 10 ⊢ (D ·P R) ∈ 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) ∈ V
56		oprex 3018	. . . . . . . 8 ⊢ (D ·P S) ∈ 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) ∈ 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)) ∈ 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)) → (((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 1131	. . 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 3018	. . . 4 ⊢ ((B ·P G) +P (D ·P R)) ∈ V
67		oprex 3018	. . . 4 ⊢ ((A ·P F) +P (C ·P S)) ∈ V
68	66, 67, 30, 31, 32	caopr13 3077	. . 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 3018	. . . 4 ⊢ ((A ·P G) +P (D ·P S)) ∈ V
70	69, 62	addasspr 3918	. . 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 1146	. 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 3078	. . 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 3010	. 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 3080	. . 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 3010	. 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 1146	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 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  (class class class)co 3001   +_P cpp 3781   ·_P cmp 3782

This theorem is referenced by:  mulcmpblnr 3977

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-mp 3883

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