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Theorem mulcmpblnr 8056
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 8055 . . 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 538 . . . . 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 539 . . . . 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 7887 . . . . 5 |- ( ( D e. P. /\ F e. P. ) -> ( D .P. F ) e. P. )
52, 3, 4syl2anc 411 . . . 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 535 . . . . . . 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 7887 . . . . . . 7 |- ( ( A e. P. /\ F e. P. ) -> ( A .P. F ) e. P. )
86, 3, 7syl2anc 411 . . . . . 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 536 . . . . . . 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 540 . . . . . . 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 7887 . . . . . . 7 |- ( ( B e. P. /\ G e. P. ) -> ( B .P. G ) e. P. )
129, 10, 11syl2anc 411 . . . . . 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 7852 . . . . . 6 |- ( ( ( A .P. F ) e. P. /\ ( B .P. G ) e. P. ) -> ( ( A .P. F ) +P. ( B .P. G ) ) e. P. )
148, 12, 13syl2anc 411 . . . . 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 537 . . . . . . 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 542 . . . . . . 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 7887 . . . . . . 7 |- ( ( C e. P. /\ S e. P. ) -> ( C .P. S ) e. P. )
1815, 16, 17syl2anc 411 . . . . . 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 541 . . . . . . 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 7887 . . . . . . 7 |- ( ( D e. P. /\ R e. P. ) -> ( D .P. R ) e. P. )
212, 19, 20syl2anc 411 . . . . . 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 7852 . . . . . 6 |- ( ( ( C .P. S ) e. P. /\ ( D .P. R ) e. P. ) -> ( ( C .P. S ) +P. ( D .P. R ) ) e. P. )
2318, 21, 22syl2anc 411 . . . . 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 7852 . . . . 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 411 . . . 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 7887 . . . . . . 7 |- ( ( A e. P. /\ G e. P. ) -> ( A .P. G ) e. P. )
276, 10, 26syl2anc 411 . . . . . 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 7887 . . . . . . 7 |- ( ( B e. P. /\ F e. P. ) -> ( B .P. F ) e. P. )
299, 3, 28syl2anc 411 . . . . . 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 7852 . . . . . 6 |- ( ( ( A .P. G ) e. P. /\ ( B .P. F ) e. P. ) -> ( ( A .P. G ) +P. ( B .P. F ) ) e. P. )
3127, 29, 30syl2anc 411 . . . . 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 7887 . . . . . . 7 |- ( ( C e. P. /\ R e. P. ) -> ( C .P. R ) e. P. )
3315, 19, 32syl2anc 411 . . . . . 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 7887 . . . . . . 7 |- ( ( D e. P. /\ S e. P. ) -> ( D .P. S ) e. P. )
352, 16, 34syl2anc 411 . . . . . 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 7852 . . . . . 6 |- ( ( ( C .P. R ) e. P. /\ ( D .P. S ) e. P. ) -> ( ( C .P. R ) +P. ( D .P. S ) ) e. P. )
3733, 35, 36syl2anc 411 . . . . 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 7852 . . . . 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 411 . . . 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 7931 . . . 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 1274 . . 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 8049 . . 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 1275 . 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 169 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 104   <-> wb 105   = wceq 1398   e. wcel 2203   <.cop 3692   class class class wbr 4109  (class class class)co 6050   P.cnp 7606   +P. cpp 7608   .P. cmp 7609   ~R cer 7611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-imp 7784  df-enr 8041
This theorem is referenced by:  mulsrmo  8059
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