Theorem mulcmpblnr 9892

Description: Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcmpblnr	|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) )

Proof of Theorem mulcmpblnr

Step	Hyp	Ref	Expression
1		mulcmpblnrlem 9891	. . 3 |- ( ( ( 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 ) ) ) ) )
2		mulclpr 9842	. . . . . 6 |- ( ( D e. P. /\ F e. P. ) -> ( D .P. F ) e. P. )
3	2	ad2ant2lr 784	. . . . 5 |- ( ( ( C e. P. /\ D e. P. ) /\ ( F e. P. /\ G e. P. ) ) -> ( D .P. F ) e. P. )
4	3	ad2ant2lr 784	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( D .P. F ) e. P. )
5		simplll 798	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> A e. P. )
6		simprll 802	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> F e. P. )
7		mulclpr 9842	. . . . . . 7 |- ( ( A e. P. /\ F e. P. ) -> ( A .P. F ) e. P. )
8	5, 6, 7	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( A .P. F ) e. P. )
9		simpllr 799	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> B e. P. )
10		simprlr 803	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> G e. P. )
11		mulclpr 9842	. . . . . . 7 |- ( ( B e. P. /\ G e. P. ) -> ( B .P. G ) e. P. )
12	9, 10, 11	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( B .P. G ) e. P. )
13		addclpr 9840	. . . . . 6 |- ( ( ( A .P. F ) e. P. /\ ( B .P. G ) e. P. ) -> ( ( A .P. F ) +P. ( B .P. G ) ) e. P. )
14	8, 12, 13	syl2anc 693	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( A .P. F ) +P. ( B .P. G ) ) e. P. )
15		simplrl 800	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> C e. P. )
16		simprrr 805	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> S e. P. )
17		mulclpr 9842	. . . . . . 7 |- ( ( C e. P. /\ S e. P. ) -> ( C .P. S ) e. P. )
18	15, 16, 17	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( C .P. S ) e. P. )
19		simplrr 801	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> D e. P. )
20		simprrl 804	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> R e. P. )
21		mulclpr 9842	. . . . . . 7 |- ( ( D e. P. /\ R e. P. ) -> ( D .P. R ) e. P. )
22	19, 20, 21	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( D .P. R ) e. P. )
23		addclpr 9840	. . . . . 6 |- ( ( ( C .P. S ) e. P. /\ ( D .P. R ) e. P. ) -> ( ( C .P. S ) +P. ( D .P. R ) ) e. P. )
24	18, 22, 23	syl2anc 693	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( C .P. S ) +P. ( D .P. R ) ) e. P. )
25		addclpr 9840	. . . . 5 |- ( ( ( ( A .P. F ) +P. ( B .P. G ) ) e. P. /\ ( ( C .P. S ) +P. ( D .P. R ) ) e. P. ) -> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) e. P. )
26	14, 24, 25	syl2anc 693	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) e. P. )
27		addcanpr 9868	. . . 4 |- ( ( ( D .P. F ) e. P. /\ ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) e. P. ) -> ( ( ( 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 ) ) ) ) -> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) = ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
28	4, 26, 27	syl2anc 693	. . 3 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( 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 ) ) ) ) -> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) = ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
29	1, 28	syl5 34	. 2 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) = ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
30		mulclpr 9842	. . . . 5 |- ( ( A e. P. /\ G e. P. ) -> ( A .P. G ) e. P. )
31		mulclpr 9842	. . . . 5 |- ( ( B e. P. /\ F e. P. ) -> ( B .P. F ) e. P. )
32		addclpr 9840	. . . . 5 |- ( ( ( A .P. G ) e. P. /\ ( B .P. F ) e. P. ) -> ( ( A .P. G ) +P. ( B .P. F ) ) e. P. )
33	30, 31, 32	syl2an 494	. . . 4 |- ( ( ( A e. P. /\ G e. P. ) /\ ( B e. P. /\ F e. P. ) ) -> ( ( A .P. G ) +P. ( B .P. F ) ) e. P. )
34	5, 10, 9, 6, 33	syl22anc 1327	. . 3 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( A .P. G ) +P. ( B .P. F ) ) e. P. )
35		mulclpr 9842	. . . . 5 |- ( ( C e. P. /\ R e. P. ) -> ( C .P. R ) e. P. )
36		mulclpr 9842	. . . . 5 |- ( ( D e. P. /\ S e. P. ) -> ( D .P. S ) e. P. )
37		addclpr 9840	. . . . 5 |- ( ( ( C .P. R ) e. P. /\ ( D .P. S ) e. P. ) -> ( ( C .P. R ) +P. ( D .P. S ) ) e. P. )
38	35, 36, 37	syl2an 494	. . . 4 |- ( ( ( C e. P. /\ R e. P. ) /\ ( D e. P. /\ S e. P. ) ) -> ( ( C .P. R ) +P. ( D .P. S ) ) e. P. )
39	15, 20, 19, 16, 38	syl22anc 1327	. . 3 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( C .P. R ) +P. ( D .P. S ) ) e. P. )
40		enrbreq 9885	. . 3 |- ( ( ( ( ( A .P. F ) +P. ( B .P. G ) ) e. P. /\ ( ( A .P. G ) +P. ( B .P. F ) ) e. P. ) /\ ( ( ( C .P. R ) +P. ( D .P. S ) ) e. P. /\ ( ( C .P. S ) +P. ( D .P. R ) ) e. P. ) ) -> ( <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. <-> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) = ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
41	14, 34, 39, 24, 40	syl22anc 1327	. 2 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. <-> ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) = ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
42	29, 41	sylibrd 249	1 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653  (class class class)co 6650   P.cnp 9681   +P. cpp 9683   .P. cmp 9684   ~R cer 9686

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-int 4476  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  df-ltp 9807  df-enr 9877

This theorem is referenced by:  mulsrmo  9895