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Theorem mulcmpblnrlemg 7660
Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
Assertion
Ref Expression
mulcmpblnrlemg  |-  ( ( ( ( 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 ) ) ) ) ) )

Proof of Theorem mulcmpblnrlemg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 524 . . . . . . . . 9  |-  ( ( ( ( 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. )
2 simprlr 528 . . . . . . . . 9  |-  ( ( ( ( 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. )
3 mulclpr 7492 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
41, 2, 3syl2anc 409 . . . . . . . 8  |-  ( ( ( ( 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. )
5 simplrr 526 . . . . . . . . 9  |-  ( ( ( ( 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. )
6 simprrl 529 . . . . . . . . 9  |-  ( ( ( ( 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. )
7 mulclpr 7492 . . . . . . . . 9  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
85, 6, 7syl2anc 409 . . . . . . . 8  |-  ( ( ( ( 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. )
9 addclpr 7457 . . . . . . . 8  |-  ( ( ( B  .P.  G
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  P. )
104, 8, 9syl2anc 409 . . . . . . 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  .P.  G )  +P.  ( D  .P.  R
) )  e.  P. )
11 simplrl 525 . . . . . . . 8  |-  ( ( ( ( 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. )
12 mulclpr 7492 . . . . . . . 8  |-  ( ( C  e.  P.  /\  G  e.  P. )  ->  ( C  .P.  G
)  e.  P. )
1311, 2, 12syl2anc 409 . . . . . . 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  .P.  G )  e.  P. )
14 simprll 527 . . . . . . . . 9  |-  ( ( ( ( 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. )
15 mulclpr 7492 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
161, 14, 15syl2anc 409 . . . . . . . 8  |-  ( ( ( ( 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. )
17 mulclpr 7492 . . . . . . . . 9  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
1811, 6, 17syl2anc 409 . . . . . . . 8  |-  ( ( ( ( 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. )
19 addclpr 7457 . . . . . . . 8  |-  ( ( ( B  .P.  F
)  e.  P.  /\  ( C  .P.  R )  e.  P. )  -> 
( ( B  .P.  F )  +P.  ( C  .P.  R ) )  e.  P. )
2016, 18, 19syl2anc 409 . . . . . . 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  .P.  F )  +P.  ( C  .P.  R
) )  e.  P. )
21 addassprg 7499 . . . . . . 7  |-  ( ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  P.  /\  ( C  .P.  G )  e. 
P.  /\  ( ( B  .P.  F )  +P.  ( C  .P.  R
) )  e.  P. )  ->  ( ( ( ( B  .P.  G
)  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G
) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  (
( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) ) )
2210, 13, 20, 21syl3anc 1220 . . . . . 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
)  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G
) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  (
( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) ) )
2322adantr 274 . . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( ( B  .P.  G )  +P.  ( D  .P.  R
) )  +P.  ( C  .P.  G ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R
) )  +P.  (
( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) ) )
24 oveq2 5832 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
2524ad2antll 483 . . . . . . . . . 10  |-  ( ( ( ( ( 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.  S ) )  =  ( D  .P.  ( G  +P.  R ) ) )
26 simprrr 530 . . . . . . . . . . . 12  |-  ( ( ( ( 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. )
27 distrprg 7508 . . . . . . . . . . . 12  |-  ( ( D  e.  P.  /\  F  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) ) )
285, 14, 26, 27syl3anc 1220 . . . . . . . . . . 11  |-  ( ( ( ( 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.  S ) )  =  ( ( D  .P.  F )  +P.  ( D  .P.  S ) ) )
2928adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( 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.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) ) )
30 distrprg 7508 . . . . . . . . . . . 12  |-  ( ( D  e.  P.  /\  G  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
315, 2, 6, 30syl3anc 1220 . . . . . . . . . . 11  |-  ( ( ( ( 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.  ( G  +P.  R ) )  =  ( ( D  .P.  G )  +P.  ( D  .P.  R ) ) )
3231adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( 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.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
3325, 29, 323eqtr3d 2198 . . . . . . . . 9  |-  ( ( ( ( ( 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.  ( D  .P.  S ) )  =  ( ( D  .P.  G )  +P.  ( D  .P.  R ) ) )
3433oveq2d 5840 . . . . . . . 8  |-  ( ( ( ( ( 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.  G
)  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S
) ) )  =  ( ( A  .P.  G )  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R ) ) ) )
35 simplll 523 . . . . . . . . . . 11  |-  ( ( ( ( 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. )
36 mulclpr 7492 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
3735, 2, 36syl2anc 409 . . . . . . . . . 10  |-  ( ( ( ( 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. )
38 mulclpr 7492 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  G  e.  P. )  ->  ( D  .P.  G
)  e.  P. )
395, 2, 38syl2anc 409 . . . . . . . . . 10  |-  ( ( ( ( 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.  G )  e.  P. )
40 addassprg 7499 . . . . . . . . . 10  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( D  .P.  G )  e.  P.  /\  ( D  .P.  R )  e. 
P. )  ->  (
( ( A  .P.  G )  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( A  .P.  G
)  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R
) ) ) )
4137, 39, 8, 40syl3anc 1220 . . . . . . . . 9  |-  ( ( ( ( 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.  ( D  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( A  .P.  G )  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R ) ) ) )
4241adantr 274 . . . . . . . 8  |-  ( ( ( ( ( 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.  G )  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( A  .P.  G
)  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R
) ) ) )
43 oveq1 5831 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
4443ad2antrl 482 . . . . . . . . . 10  |-  ( ( ( ( ( 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.  D
)  .P.  G )  =  ( ( B  +P.  C )  .P. 
G ) )
45 distrprg 7508 . . . . . . . . . . . . 13  |-  ( ( G  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) ) )
462, 35, 5, 45syl3anc 1220 . . . . . . . . . . . 12  |-  ( ( ( ( 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  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A )  +P.  ( G  .P.  D ) ) )
47 addclpr 7457 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  D  e.  P. )  ->  ( A  +P.  D
)  e.  P. )
4835, 5, 47syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )  e.  P. )
49 mulcomprg 7500 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  D
)  e.  P.  /\  G  e.  P. )  ->  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) ) )
5048, 2, 49syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )  .P. 
G )  =  ( G  .P.  ( A  +P.  D ) ) )
51 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  =  ( G  .P.  A ) )
5235, 2, 51syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )  =  ( G  .P.  A ) )
53 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( D  e.  P.  /\  G  e.  P. )  ->  ( D  .P.  G
)  =  ( G  .P.  D ) )
545, 2, 53syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  G )  =  ( G  .P.  D ) )
5552, 54oveq12d 5842 . . . . . . . . . . . 12  |-  ( ( ( ( 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.  ( D  .P.  G
) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) ) )
5646, 50, 553eqtr4d 2200 . . . . . . . . . . 11  |-  ( ( ( ( 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 )  .P. 
G )  =  ( ( A  .P.  G
)  +P.  ( D  .P.  G ) ) )
5756adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( 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.  D
)  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G
) ) )
58 distrprg 7508 . . . . . . . . . . . . 13  |-  ( ( G  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) ) )
592, 1, 11, 58syl3anc 1220 . . . . . . . . . . . 12  |-  ( ( ( ( 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  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B )  +P.  ( G  .P.  C ) ) )
60 addclpr 7457 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
611, 11, 60syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  C )  e.  P. )
62 mulcomprg 7500 . . . . . . . . . . . . 13  |-  ( ( ( B  +P.  C
)  e.  P.  /\  G  e.  P. )  ->  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) ) )
6361, 2, 62syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( 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.  C )  .P. 
G )  =  ( G  .P.  ( B  +P.  C ) ) )
64 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  =  ( G  .P.  B ) )
651, 2, 64syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )  =  ( G  .P.  B ) )
66 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  G  e.  P. )  ->  ( C  .P.  G
)  =  ( G  .P.  C ) )
6711, 2, 66syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  G )  =  ( G  .P.  C ) )
6865, 67oveq12d 5842 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )  +P.  ( C  .P.  G
) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) ) )
6959, 63, 683eqtr4d 2200 . . . . . . . . . . 11  |-  ( ( ( ( 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.  C )  .P. 
G )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
7069adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) ) )  ->  (
( B  +P.  C
)  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G
) ) )
7144, 57, 703eqtr3d 2198 . . . . . . . . 9  |-  ( ( ( ( ( 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.  G
)  +P.  ( D  .P.  G ) )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) ) )
7271oveq1d 5839 . . . . . . . 8  |-  ( ( ( ( ( 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.  G )  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  +P.  ( D  .P.  R ) ) )
7334, 42, 723eqtr2d 2196 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( A  .P.  G
)  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S
) ) )  =  ( ( ( B  .P.  G )  +P.  ( C  .P.  G
) )  +P.  ( D  .P.  R ) ) )
74 mulclpr 7492 . . . . . . . . . 10  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
755, 14, 74syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( 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. )
76 mulclpr 7492 . . . . . . . . . 10  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
775, 26, 76syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( 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. )
78 addcomprg 7498 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
7978adantl 275 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  /\  ( x  e. 
P.  /\  y  e.  P. ) )  ->  (
x  +P.  y )  =  ( y  +P.  x ) )
80 addassprg 7499 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
( x  +P.  y
)  +P.  z )  =  ( x  +P.  ( y  +P.  z
) ) )
8180adantl 275 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  /\  ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. ) )  ->  (
( x  +P.  y
)  +P.  z )  =  ( x  +P.  ( y  +P.  z
) ) )
8237, 75, 77, 79, 81caov12d 6002 . . . . . . . 8  |-  ( ( ( ( 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.  ( ( D  .P.  F )  +P.  ( D  .P.  S ) ) )  =  ( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) ) )
8382adantr 274 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( A  .P.  G
)  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S
) ) )  =  ( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) ) )
844, 13, 8, 79, 81caov32d 6001 . . . . . . . 8  |-  ( ( ( ( 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 )  +P.  ( C  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) ) )
8584adantr 274 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( B  .P.  G )  +P.  ( C  .P.  G ) )  +P.  ( D  .P.  R ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) ) )
8673, 83, 853eqtr3d 2198 . . . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( D  .P.  F
)  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R
) )  +P.  ( C  .P.  G ) ) )
8786oveq1d 5839 . . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) )
88 oveq1 5831 . . . . . . . . . . . 12  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
8988adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
90 distrprg 7508 . . . . . . . . . . . . . 14  |-  ( ( F  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) ) )
9114, 35, 5, 90syl3anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( ( 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  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A )  +P.  ( F  .P.  D ) ) )
92 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( ( A  +P.  D
)  e.  P.  /\  F  e.  P. )  ->  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) ) )
9348, 14, 92syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )  .P. 
F )  =  ( F  .P.  ( A  +P.  D ) ) )
94 mulcomprg 7500 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  =  ( F  .P.  A ) )
9535, 14, 94syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )  =  ( F  .P.  A ) )
96 mulcomprg 7500 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  =  ( F  .P.  D ) )
975, 14, 96syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )  =  ( F  .P.  D ) )
9895, 97oveq12d 5842 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  ( D  .P.  F
) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) ) )
9991, 93, 983eqtr4d 2200 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )  .P. 
F )  =  ( ( A  .P.  F
)  +P.  ( D  .P.  F ) ) )
10099adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( A  .P.  F
)  +P.  ( D  .P.  F ) ) )
101 distrprg 7508 . . . . . . . . . . . . . 14  |-  ( ( F  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) ) )
10214, 1, 11, 101syl3anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( ( 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  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B )  +P.  ( F  .P.  C ) ) )
103 mulcomprg 7500 . . . . . . . . . . . . . 14  |-  ( ( ( B  +P.  C
)  e.  P.  /\  F  e.  P. )  ->  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) ) )
10461, 14, 103syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  C )  .P. 
F )  =  ( F  .P.  ( B  +P.  C ) ) )
105 mulcomprg 7500 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  =  ( F  .P.  B ) )
1061, 14, 105syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )  =  ( F  .P.  B ) )
107 mulcomprg 7500 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  P.  /\  F  e.  P. )  ->  ( C  .P.  F
)  =  ( F  .P.  C ) )
10811, 14, 107syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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.  F )  =  ( F  .P.  C ) )
109106, 108oveq12d 5842 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )  +P.  ( C  .P.  F
) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) ) )
110102, 104, 1093eqtr4d 2200 . . . . . . . . . . . 12  |-  ( ( ( ( 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.  C )  .P. 
F )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
111110adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) )  ->  ( ( B  +P.  C )  .P. 
F )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
11289, 100, 1113eqtr3d 2198 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
113112oveq1d 5839 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  ->  ( ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) ) )
114113adantrr 471 . . . . . . . 8  |-  ( ( ( ( ( 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.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) ) )
115 mulclpr 7492 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  F  e.  P. )  ->  ( C  .P.  F
)  e.  P. )
11611, 14, 115syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( 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.  F )  e.  P. )
117 mulclpr 7492 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
11811, 26, 117syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( 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. )
119 addassprg 7499 . . . . . . . . . . . 12  |-  ( ( ( B  .P.  F
)  e.  P.  /\  ( C  .P.  F )  e.  P.  /\  ( C  .P.  S )  e. 
P. )  ->  (
( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S
) ) ) )
12016, 116, 118, 119syl3anc 1220 . . . . . . . . . . 11  |-  ( ( ( ( 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 )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F )  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) ) )
121120adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F )  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) ) )
122 oveq2 5832 . . . . . . . . . . . . 13  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
123122adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( C  .P.  ( F  +P.  S ) )  =  ( C  .P.  ( G  +P.  R ) ) )
124 distrprg 7508 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  F  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) ) )
12511, 14, 26, 124syl3anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  ( F  +P.  S ) )  =  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) )
126125adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) )
127 distrprg 7508 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  G  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
12811, 2, 6, 127syl3anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( ( 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.  ( G  +P.  R ) )  =  ( ( C  .P.  G )  +P.  ( C  .P.  R ) ) )
129128adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G )  +P.  ( C  .P.  R ) ) )
130123, 126, 1293eqtr3d 2198 . . . . . . . . . . 11  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
131130oveq2d 5840 . . . . . . . . . 10  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( ( B  .P.  F )  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) )  =  ( ( B  .P.  F )  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
132121, 131eqtrd 2190 . . . . . . . . 9  |-  ( ( ( ( ( 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  +P.  S )  =  ( G  +P.  R ) )  ->  ( ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F )  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R ) ) ) )
133132adantrl 470 . . . . . . . 8  |-  ( ( ( ( ( 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 ) ) )  ->  (
( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
134114, 133eqtrd 2190 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( A  .P.  F )  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
135 mulclpr 7492 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
13635, 14, 135syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( 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. )
137136, 75, 118, 79, 81caov32d 6001 . . . . . . . 8  |-  ( ( ( ( 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.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )
138137adantr 274 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( A  .P.  F )  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )
13916, 13, 18, 79, 81caov12d 6002 . . . . . . . 8  |-  ( ( ( ( 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 )  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R ) ) )  =  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) )
140139adantr 274 . . . . . . 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  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( B  .P.  F
)  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) )  =  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) )
141134, 138, 1403eqtr3d 2198 . . . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) )  =  ( ( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) )
142141oveq2d 5840 . . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( ( B  .P.  G
)  +P.  ( D  .P.  R ) )  +P.  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) ) )
14323, 87, 1423eqtr4rd 2201 . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( ( D  .P.  F
)  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) )
144 addclpr 7457 . . . . . . 7  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( C  .P.  S )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  P. )
145136, 118, 144syl2anc 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 )  +P.  ( C  .P.  S
) )  e.  P. )
14610, 145, 75, 79, 81caov13d 6004 . . . . 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. ) ) )  ->  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  (
( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F
)  +P.  ( C  .P.  S ) )  +P.  ( ( B  .P.  G )  +P.  ( D  .P.  R ) ) ) ) )
147146adantr 274 . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  (
( B  .P.  G
)  +P.  ( D  .P.  R ) ) ) ) )
148 addclpr 7457 . . . . . . 7  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( D  .P.  S ) )  e.  P. )
14937, 77, 148syl2anc 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 )  +P.  ( D  .P.  S
) )  e.  P. )
150 addassprg 7499 . . . . . 6  |-  ( ( ( D  .P.  F
)  e.  P.  /\  ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  e.  P.  /\  (
( B  .P.  F
)  +P.  ( C  .P.  R ) )  e. 
P. )  ->  (
( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) ) )
15175, 149, 20, 150syl3anc 1220 . . . . 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  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) ) )
152151adantr 274 . . . 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.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  (
( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) ) )
153143, 147, 1523eqtr3d 2198 . . 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.  ( C  .P.  S ) )  +P.  ( ( B  .P.  G )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  G
)  +P.  ( D  .P.  S ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) ) )
154 addclpr 7457 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
155154adantl 275 . . . . . 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. ) ) )  /\  ( x  e. 
P.  /\  y  e.  P. ) )  ->  (
x  +P.  y )  e.  P. )
156136, 118, 4, 79, 81, 8, 155caov4d 6005 . . . . 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.  ( C  .P.  S ) )  +P.  (
( B  .P.  G
)  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) ) )
157156oveq2d 5840 . . . 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 )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S
) )  +P.  (
( B  .P.  G
)  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) ) ) )
158157adantr 274 . . 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.  ( C  .P.  S ) )  +P.  ( ( B  .P.  G )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) ) ) )
15937, 77, 16, 79, 81, 18, 155caov42d 6007 . . . . 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.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) )
160159oveq2d 5840 . . . 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 )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S
) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) )
161160adantr 274 . . 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.  G
)  +P.  ( D  .P.  S ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) ) ) )
162153, 158, 1613eqtr3d 2198 . 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 ) ) )  ->  (
( 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 ) ) ) ) )
163162ex 114 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
) )  ->  (
( 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 ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128  (class class class)co 5824   P.cnp 7211    +P. cpp 7213    .P. cmp 7214
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 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
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 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-imp 7389
This theorem is referenced by:  mulcmpblnr  7661
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