Theorem mulcmpblnrlem 9891

Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)

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

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . . . . 9 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A +P. D ) .P. F ) = ( ( B +P. C ) .P. F ) )
2		distrpr 9850	. . . . . . . . . 10 |- ( F .P. ( A +P. D ) ) = ( ( F .P. A ) +P. ( F .P. D ) )
3		mulcompr 9845	. . . . . . . . . 10 |- ( ( A +P. D ) .P. F ) = ( F .P. ( A +P. D ) )
4		mulcompr 9845	. . . . . . . . . . 11 |- ( A .P. F ) = ( F .P. A )
5		mulcompr 9845	. . . . . . . . . . 11 |- ( D .P. F ) = ( F .P. D )
6	4, 5	oveq12i 6662	. . . . . . . . . 10 |- ( ( A .P. F ) +P. ( D .P. F ) ) = ( ( F .P. A ) +P. ( F .P. D ) )
7	2, 3, 6	3eqtr4i 2654	. . . . . . . . 9 |- ( ( A +P. D ) .P. F ) = ( ( A .P. F ) +P. ( D .P. F ) )
8		distrpr 9850	. . . . . . . . . 10 |- ( F .P. ( B +P. C ) ) = ( ( F .P. B ) +P. ( F .P. C ) )
9		mulcompr 9845	. . . . . . . . . 10 |- ( ( B +P. C ) .P. F ) = ( F .P. ( B +P. C ) )
10		mulcompr 9845	. . . . . . . . . . 11 |- ( B .P. F ) = ( F .P. B )
11		mulcompr 9845	. . . . . . . . . . 11 |- ( C .P. F ) = ( F .P. C )
12	10, 11	oveq12i 6662	. . . . . . . . . 10 |- ( ( B .P. F ) +P. ( C .P. F ) ) = ( ( F .P. B ) +P. ( F .P. C ) )
13	8, 9, 12	3eqtr4i 2654	. . . . . . . . 9 |- ( ( B +P. C ) .P. F ) = ( ( B .P. F ) +P. ( C .P. F ) )
14	1, 7, 13	3eqtr3g 2679	. . . . . . . 8 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A .P. F ) +P. ( D .P. F ) ) = ( ( B .P. F ) +P. ( C .P. F ) ) )
15	14	oveq1d 6665	. . . . . . 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		addasspr 9844	. . . . . . . 8 |- ( ( ( B .P. F ) +P. ( C .P. F ) ) +P. ( C .P. S ) ) = ( ( B .P. F ) +P. ( ( C .P. F ) +P. ( C .P. S ) ) )
17		oveq2 6658	. . . . . . . . . 10 |- ( ( F +P. S ) = ( G +P. R ) -> ( C .P. ( F +P. S ) ) = ( C .P. ( G +P. R ) ) )
18		distrpr 9850	. . . . . . . . . 10 |- ( C .P. ( F +P. S ) ) = ( ( C .P. F ) +P. ( C .P. S ) )
19		distrpr 9850	. . . . . . . . . 10 |- ( C .P. ( G +P. R ) ) = ( ( C .P. G ) +P. ( C .P. R ) )
20	17, 18, 19	3eqtr3g 2679	. . . . . . . . 9 |- ( ( F +P. S ) = ( G +P. R ) -> ( ( C .P. F ) +P. ( C .P. S ) ) = ( ( C .P. G ) +P. ( C .P. R ) ) )
21	20	oveq2d 6666	. . . . . . . 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 ) ) ) )
22	16, 21	syl5eq 2668	. . . . . . 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 ) ) ) )
23	15, 22	sylan9eq 2676	. . . . . 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 ) ) ) )
24		ovex 6678	. . . . . . 7 |- ( A .P. F ) e. _V
25		ovex 6678	. . . . . . 7 |- ( D .P. F ) e. _V
26		ovex 6678	. . . . . . 7 |- ( C .P. S ) e. _V
27		addcompr 9843	. . . . . . 7 |- ( x +P. y ) = ( y +P. x )
28		addasspr 9844	. . . . . . 7 |- ( ( x +P. y ) +P. z ) = ( x +P. ( y +P. z ) )
29	24, 25, 26, 27, 28	caov32 6861	. . . . . 6 |- ( ( ( A .P. F ) +P. ( D .P. F ) ) +P. ( C .P. S ) ) = ( ( ( A .P. F ) +P. ( C .P. S ) ) +P. ( D .P. F ) )
30		ovex 6678	. . . . . . 7 |- ( B .P. F ) e. _V
31		ovex 6678	. . . . . . 7 |- ( C .P. G ) e. _V
32		ovex 6678	. . . . . . 7 |- ( C .P. R ) e. _V
33	30, 31, 32, 27, 28	caov12 6862	. . . . . 6 |- ( ( B .P. F ) +P. ( ( C .P. G ) +P. ( C .P. R ) ) ) = ( ( C .P. G ) +P. ( ( B .P. F ) +P. ( C .P. R ) ) )
34	23, 29, 33	3eqtr3g 2679	. . . . 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 ) ) ) )
35	34	oveq2d 6666	. . . 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 ) ) ) ) )
36		oveq2 6658	. . . . . . . . . . 11 |- ( ( F +P. S ) = ( G +P. R ) -> ( D .P. ( F +P. S ) ) = ( D .P. ( G +P. R ) ) )
37		distrpr 9850	. . . . . . . . . . 11 |- ( D .P. ( F +P. S ) ) = ( ( D .P. F ) +P. ( D .P. S ) )
38		distrpr 9850	. . . . . . . . . . 11 |- ( D .P. ( G +P. R ) ) = ( ( D .P. G ) +P. ( D .P. R ) )
39	36, 37, 38	3eqtr3g 2679	. . . . . . . . . 10 |- ( ( F +P. S ) = ( G +P. R ) -> ( ( D .P. F ) +P. ( D .P. S ) ) = ( ( D .P. G ) +P. ( D .P. R ) ) )
40	39	oveq2d 6666	. . . . . . . . 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 ) ) ) )
41		addasspr 9844	. . . . . . . . 9 |- ( ( ( A .P. G ) +P. ( D .P. G ) ) +P. ( D .P. R ) ) = ( ( A .P. G ) +P. ( ( D .P. G ) +P. ( D .P. R ) ) )
42	40, 41	syl6eqr 2674	. . . . . . . 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 ) ) )
43		oveq1 6657	. . . . . . . . . 10 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A +P. D ) .P. G ) = ( ( B +P. C ) .P. G ) )
44		distrpr 9850	. . . . . . . . . . 11 |- ( G .P. ( A +P. D ) ) = ( ( G .P. A ) +P. ( G .P. D ) )
45		mulcompr 9845	. . . . . . . . . . 11 |- ( ( A +P. D ) .P. G ) = ( G .P. ( A +P. D ) )
46		mulcompr 9845	. . . . . . . . . . . 12 |- ( A .P. G ) = ( G .P. A )
47		mulcompr 9845	. . . . . . . . . . . 12 |- ( D .P. G ) = ( G .P. D )
48	46, 47	oveq12i 6662	. . . . . . . . . . 11 |- ( ( A .P. G ) +P. ( D .P. G ) ) = ( ( G .P. A ) +P. ( G .P. D ) )
49	44, 45, 48	3eqtr4i 2654	. . . . . . . . . 10 |- ( ( A +P. D ) .P. G ) = ( ( A .P. G ) +P. ( D .P. G ) )
50		distrpr 9850	. . . . . . . . . . 11 |- ( G .P. ( B +P. C ) ) = ( ( G .P. B ) +P. ( G .P. C ) )
51		mulcompr 9845	. . . . . . . . . . 11 |- ( ( B +P. C ) .P. G ) = ( G .P. ( B +P. C ) )
52		mulcompr 9845	. . . . . . . . . . . 12 |- ( B .P. G ) = ( G .P. B )
53		mulcompr 9845	. . . . . . . . . . . 12 |- ( C .P. G ) = ( G .P. C )
54	52, 53	oveq12i 6662	. . . . . . . . . . 11 |- ( ( B .P. G ) +P. ( C .P. G ) ) = ( ( G .P. B ) +P. ( G .P. C ) )
55	50, 51, 54	3eqtr4i 2654	. . . . . . . . . 10 |- ( ( B +P. C ) .P. G ) = ( ( B .P. G ) +P. ( C .P. G ) )
56	43, 49, 55	3eqtr3g 2679	. . . . . . . . 9 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A .P. G ) +P. ( D .P. G ) ) = ( ( B .P. G ) +P. ( C .P. G ) ) )
57	56	oveq1d 6665	. . . . . . . 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 ) ) )
58	42, 57	sylan9eqr 2678	. . . . . . 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 ) ) )
59		ovex 6678	. . . . . . . 8 |- ( A .P. G ) e. _V
60		ovex 6678	. . . . . . . 8 |- ( D .P. S ) e. _V
61	59, 25, 60, 27, 28	caov12 6862	. . . . . . 7 |- ( ( A .P. G ) +P. ( ( D .P. F ) +P. ( D .P. S ) ) ) = ( ( D .P. F ) +P. ( ( A .P. G ) +P. ( D .P. S ) ) )
62		ovex 6678	. . . . . . . 8 |- ( B .P. G ) e. _V
63		ovex 6678	. . . . . . . 8 |- ( D .P. R ) e. _V
64	62, 31, 63, 27, 28	caov32 6861	. . . . . . 7 |- ( ( ( B .P. G ) +P. ( C .P. G ) ) +P. ( D .P. R ) ) = ( ( ( B .P. G ) +P. ( D .P. R ) ) +P. ( C .P. G ) )
65	58, 61, 64	3eqtr3g 2679	. . . . . 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 ) ) )
66	65	oveq1d 6665	. . . . 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 ) ) ) )
67		addasspr 9844	. . . . 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 ) ) ) )
68	66, 67	syl6eq 2672	. . . 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 ) ) ) ) )
69	35, 68	eqtr4d 2659	. . 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 ) ) ) )
70		ovex 6678	. . . 4 |- ( ( B .P. G ) +P. ( D .P. R ) ) e. _V
71		ovex 6678	. . . 4 |- ( ( A .P. F ) +P. ( C .P. S ) ) e. _V
72	70, 71, 25, 27, 28	caov13 6864	. . 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 ) ) ) )
73		addasspr 9844	. . 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 ) ) ) )
74	69, 72, 73	3eqtr3g 2679	. 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 ) ) ) ) )
75	24, 26, 62, 27, 28, 63	caov4 6865	. . 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 ) ) )
76	75	oveq2i 6661	. 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 ) ) ) )
77	59, 60, 30, 27, 28, 32	caov42 6867	. . 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 ) ) )
78	77	oveq2i 6661	. 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 ) ) ) )
79	74, 76, 78	3eqtr3g 2679	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 4   /\ wa 384   = wceq 1483  (class class class)co 6650   +P. cpp 9683   .P. cmp 9684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-mp 9806

This theorem is referenced by:  mulcmpblnr  9892