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Theorem mulcmpblnr 7753
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 7752 . . 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 7584 . . . . 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 7584 . . . . . . 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 7584 . . . . . . 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 7549 . . . . . 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 7584 . . . . . . 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 7584 . . . . . . 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 7549 . . . . . 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 7549 . . . . 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 7584 . . . . . . 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 7584 . . . . . . 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 7549 . . . . . 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 7584 . . . . . . 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 7584 . . . . . . 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 7549 . . . . . 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 7549 . . . . 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 7628 . . . 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 1248 . . 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 7746 . . 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 1249 . 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 1363    e. wcel 2158   <.cop 3607   class class class wbr 4015  (class class class)co 5888   P.cnp 7303    +P. cpp 7305    .P. cmp 7306    ~R cer 7308
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-iplp 7480  df-imp 7481  df-enr 7738
This theorem is referenced by:  mulsrmo  7756
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