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Theorem mulcmpblnr 7874
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 7873 . . 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 536 . . . . 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 537 . . . . 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 7705 . . . . 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 533 . . . . . . 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 7705 . . . . . . 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 534 . . . . . . 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 538 . . . . . . 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 7705 . . . . . . 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 7670 . . . . . 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 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. ) ) ) -> C e. P. )
16 simprrr 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. ) ) ) -> S e. P. )
17 mulclpr 7705 . . . . . . 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 539 . . . . . . 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 7705 . . . . . . 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 7670 . . . . . 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 7670 . . . . 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 7705 . . . . . . 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 7705 . . . . . . 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 7670 . . . . . 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 7705 . . . . . . 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 7705 . . . . . . 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 7670 . . . . . 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 7670 . . . . 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 7749 . . . 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 1250 . . 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 7867 . . 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 1251 . 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 1373   e. wcel 2177   <.cop 3641   class class class wbr 4051  (class class class)co 5957   P.cnp 7424   +P. cpp 7426   .P. cmp 7427   ~R cer 7429
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-2o 6516  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486  df-enq0 7557  df-nq0 7558  df-0nq0 7559  df-plq0 7560  df-mq0 7561  df-inp 7599  df-iplp 7601  df-imp 7602  df-enr 7859
This theorem is referenced by:  mulsrmo  7877
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