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Theorem mulcmpblnr 9460
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
StepHypRef Expression
1 mulcmpblnrlem 9459 . . 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 9410 . . . . . 6  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
32ad2ant2lr 747 . . . . 5  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( D  .P.  F )  e.  P. )
43ad2ant2lr 747 . . . 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 757 . . . . . . 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 761 . . . . . . 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 9410 . . . . . . 7  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
85, 6, 7syl2anc 661 . . . . . 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 758 . . . . . . 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 762 . . . . . . 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 9410 . . . . . . 7  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
129, 10, 11syl2anc 661 . . . . . 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 9408 . . . . . 6  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( B  .P.  G )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P. )
148, 12, 13syl2anc 661 . . . . 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 759 . . . . . . 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 764 . . . . . . 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 9410 . . . . . . 7  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
1815, 16, 17syl2anc 661 . . . . . 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 760 . . . . . . 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 763 . . . . . . 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 9410 . . . . . . 7  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
2219, 20, 21syl2anc 661 . . . . . 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 9408 . . . . . 6  |-  ( ( ( C  .P.  S
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. )
2418, 22, 23syl2anc 661 . . . . 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 9408 . . . . 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. )
2614, 24, 25syl2anc 661 . . . 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 9436 . . . 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 ) ) ) ) )
284, 26, 27syl2anc 661 . . 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 ) ) ) ) )
291, 28syl5 32 . 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 9410 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
31 mulclpr 9410 . . . . 5  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
32 addclpr 9408 . . . . 5  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( B  .P.  F )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )
3330, 31, 32syl2an 477 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. )
)  ->  ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  e.  P. )
345, 10, 9, 6, 33syl22anc 1229 . . 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 9410 . . . . 5  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
36 mulclpr 9410 . . . . 5  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
37 addclpr 9408 . . . . 5  |-  ( ( ( C  .P.  R
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( C  .P.  R )  +P.  ( D  .P.  S ) )  e.  P. )
3835, 36, 37syl2an 477 . . . 4  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P. )
3915, 20, 19, 16, 38syl22anc 1229 . . 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 9453 . . 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 ) ) ) ) )
4114, 34, 39, 24, 40syl22anc 1229 . 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 ) ) ) ) )
4229, 41sylibrd 234 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453  (class class class)co 6295   P.cnp 9249    +P. cpp 9251    .P. cmp 9252    ~R cer 9254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ni 9262  df-pli 9263  df-mi 9264  df-lti 9265  df-plpq 9298  df-mpq 9299  df-ltpq 9300  df-enq 9301  df-nq 9302  df-erq 9303  df-plq 9304  df-mq 9305  df-1nq 9306  df-rq 9307  df-ltnq 9308  df-np 9371  df-plp 9373  df-mp 9374  df-ltp 9375  df-enr 9445
This theorem is referenced by:  mulsrmo  9463
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