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Theorem mulcmpblnr 7662
Description: Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
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
StepHypRef Expression
1 mulcmpblnrlemg 7661 . . 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. 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 simplrr 526 . . . . 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. ) ) ) -> D e. P. )
3 simprll 527 . . . . 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. ) ) ) -> F e. P. )
4 mulclpr 7493 . . . . 5 |- ( ( D e. P. /\ F e. P. ) -> ( D .P. F ) e. P. )
52, 3, 4syl2anc 409 . . . 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. )
6 simplll 523 . . . . . . 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. )
7 mulclpr 7493 . . . . . . 7 |- ( ( A e. P. /\ F e. P. ) -> ( A .P. F ) e. P. )
86, 3, 7syl2anc 409 . . . . . 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 524 . . . . . . 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 528 . . . . . . 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 7493 . . . . . . 7 |- ( ( B e. P. /\ G e. P. ) -> ( B .P. G ) e. P. )
129, 10, 11syl2anc 409 . . . . . 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 7458 . . . . . 6 |- ( ( ( A .P. F ) e. P. /\ ( B .P. G ) e. P. ) -> ( ( A .P. F ) +P. ( B .P. G ) ) e. P. )
148, 12, 13syl2anc 409 . . . . 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 525 . . . . . . 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 530 . . . . . . 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 7493 . . . . . . 7 |- ( ( C e. P. /\ S e. P. ) -> ( C .P. S ) e. P. )
1815, 16, 17syl2anc 409 . . . . . 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 simprrl 529 . . . . . . 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. )
20 mulclpr 7493 . . . . . . 7 |- ( ( D e. P. /\ R e. P. ) -> ( D .P. R ) e. P. )
212, 19, 20syl2anc 409 . . . . . 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. )
22 addclpr 7458 . . . . . 6 |- ( ( ( C .P. S ) e. P. /\ ( D .P. R ) e. P. ) -> ( ( C .P. S ) +P. ( D .P. R ) ) e. P. )
2318, 21, 22syl2anc 409 . . . . 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. )
24 addclpr 7458 . . . . 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. )
2514, 23, 24syl2anc 409 . . . 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. )
26 mulclpr 7493 . . . . . . 7 |- ( ( A e. P. /\ G e. P. ) -> ( A .P. G ) e. P. )
276, 10, 26syl2anc 409 . . . . . 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. G ) e. P. )
28 mulclpr 7493 . . . . . . 7 |- ( ( B e. P. /\ F e. P. ) -> ( B .P. F ) e. P. )
299, 3, 28syl2anc 409 . . . . . 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. F ) e. P. )
30 addclpr 7458 . . . . . 6 |- ( ( ( A .P. G ) e. P. /\ ( B .P. F ) e. P. ) -> ( ( A .P. G ) +P. ( B .P. F ) ) e. P. )
3127, 29, 30syl2anc 409 . . . . 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. G ) +P. ( B .P. F ) ) e. P. )
32 mulclpr 7493 . . . . . . 7 |- ( ( C e. P. /\ R e. P. ) -> ( C .P. R ) e. P. )
3315, 19, 32syl2anc 409 . . . . . 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. R ) e. P. )
34 mulclpr 7493 . . . . . . 7 |- ( ( D e. P. /\ S e. P. ) -> ( D .P. S ) e. P. )
352, 16, 34syl2anc 409 . . . . . 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. S ) e. P. )
36 addclpr 7458 . . . . . 6 |- ( ( ( C .P. R ) e. P. /\ ( D .P. S ) e. P. ) -> ( ( C .P. R ) +P. ( D .P. S ) ) e. P. )
3733, 35, 36syl2anc 409 . . . . 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. R ) +P. ( D .P. S ) ) e. P. )
38 addclpr 7458 . . . . 5 |- ( ( ( ( A .P. G ) +P. ( B .P. F ) ) e. P. /\ ( ( C .P. R ) +P. ( D .P. S ) ) e. P. ) -> ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) e. P. )
3931, 37, 38syl2anc 409 . . . 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. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) e. P. )
40 addcanprg 7537 . . . 4 |- ( ( ( D .P. F ) e. P. /\ ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) e. P. /\ ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .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 ) ) ) ) )
415, 25, 39, 40syl3anc 1220 . . 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 ) ) ) ) )
421, 41syld 45 . 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 ) ) ) ) )
43 enrbreq 7655 . . 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 ) ) ) ) )
4414, 31, 37, 23, 43syl22anc 1221 . 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 ) ) ) ) )
4542, 44sylibrd 168 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 ) ) >. ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   <.cop 3563   class class class wbr 3966  (class class class)co 5825   P.cnp 7212   +P. cpp 7214   .P. cmp 7215   ~R cer 7217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-eprel 4250  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-1o 6364  df-2o 6365  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-pli 7226  df-mi 7227  df-lti 7228  df-plpq 7265  df-mpq 7266  df-enq 7268  df-nqqs 7269  df-plqqs 7270  df-mqqs 7271  df-1nqqs 7272  df-rq 7273  df-ltnqqs 7274  df-enq0 7345  df-nq0 7346  df-0nq0 7347  df-plq0 7348  df-mq0 7349  df-inp 7387  df-iplp 7389  df-imp 7390  df-enr 7647
This theorem is referenced by:  mulsrmo  7665
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