ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulcmpblnrlemg Unicode version

Theorem mulcmpblnrlemg 6979
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 501 . . . . . . . . 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 505 . . . . . . . . 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 6824 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
41, 2, 3syl2anc 403 . . . . . . . 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 503 . . . . . . . . 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 506 . . . . . . . . 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 6824 . . . . . . . . 9  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
85, 6, 7syl2anc 403 . . . . . . . 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 6789 . . . . . . . 8  |-  ( ( ( B  .P.  G
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  P. )
104, 8, 9syl2anc 403 . . . . . . 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 502 . . . . . . . 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 6824 . . . . . . . 8  |-  ( ( C  e.  P.  /\  G  e.  P. )  ->  ( C  .P.  G
)  e.  P. )
1311, 2, 12syl2anc 403 . . . . . . 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 504 . . . . . . . . 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 6824 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
161, 14, 15syl2anc 403 . . . . . . . 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 6824 . . . . . . . . 9  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
1811, 6, 17syl2anc 403 . . . . . . . 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 6789 . . . . . . . 8  |-  ( ( ( B  .P.  F
)  e.  P.  /\  ( C  .P.  R )  e.  P. )  -> 
( ( B  .P.  F )  +P.  ( C  .P.  R ) )  e.  P. )
2016, 18, 19syl2anc 403 . . . . . . 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 6831 . . . . . . 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 1170 . . . . . 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 270 . . . . 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 5551 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
2524ad2antll 475 . . . . . . . . . 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 507 . . . . . . . . . . . 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 6840 . . . . . . . . . . . 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 1170 . . . . . . . . . . 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 270 . . . . . . . . . 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 6840 . . . . . . . . . . . 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 1170 . . . . . . . . . . 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 270 . . . . . . . . . 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 2122 . . . . . . . . 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 5559 . . . . . . . 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 500 . . . . . . . . . . 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 6824 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
3735, 2, 36syl2anc 403 . . . . . . . . . 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 6824 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  G  e.  P. )  ->  ( D  .P.  G
)  e.  P. )
395, 2, 38syl2anc 403 . . . . . . . . . 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 6831 . . . . . . . . . 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 1170 . . . . . . . . 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 270 . . . . . . . 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 5550 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
4443ad2antrl 474 . . . . . . . . . 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 6840 . . . . . . . . . . . . 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 1170 . . . . . . . . . . . 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 6789 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  D  e.  P. )  ->  ( A  +P.  D
)  e.  P. )
4835, 5, 47syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  D
)  e.  P.  /\  G  e.  P. )  ->  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) ) )
5048, 2, 49syl2anc 403 . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  =  ( G  .P.  A ) )
5235, 2, 51syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( D  e.  P.  /\  G  e.  P. )  ->  ( D  .P.  G
)  =  ( G  .P.  D ) )
545, 2, 53syl2anc 403 . . . . . . . . . . . . 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 5561 . . . . . . . . . . . 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 2124 . . . . . . . . . . 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 270 . . . . . . . . . 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 6840 . . . . . . . . . . . . 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 1170 . . . . . . . . . . . 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 6789 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
611, 11, 60syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . 13  |-  ( ( ( B  +P.  C
)  e.  P.  /\  G  e.  P. )  ->  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) ) )
6361, 2, 62syl2anc 403 . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  =  ( G  .P.  B ) )
651, 2, 64syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  G  e.  P. )  ->  ( C  .P.  G
)  =  ( G  .P.  C ) )
6711, 2, 66syl2anc 403 . . . . . . . . . . . . 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 5561 . . . . . . . . . . . 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 2124 . . . . . . . . . . 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 270 . . . . . . . . . 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 2122 . . . . . . . . 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 5558 . . . . . . . 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 2120 . . . . . . 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 6824 . . . . . . . . . 10  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
755, 14, 74syl2anc 403 . . . . . . . . 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 6824 . . . . . . . . . 10  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
775, 26, 76syl2anc 403 . . . . . . . . 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 6830 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
7978adantl 271 . . . . . . . . 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 6831 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
( x  +P.  y
)  +P.  z )  =  ( x  +P.  ( y  +P.  z
) ) )
8180adantl 271 . . . . . . . . 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 5713 . . . . . . . 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 270 . . . . . . 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 5712 . . . . . . . 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 270 . . . . . . 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 2122 . . . . . 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 5558 . . . . 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 5550 . . . . . . . . . . . 12  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
8988adantl 271 . . . . . . . . . . 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 6840 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( ( A  +P.  D
)  e.  P.  /\  F  e.  P. )  ->  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) ) )
9348, 14, 92syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  =  ( F  .P.  A ) )
9535, 14, 94syl2anc 403 . . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  =  ( F  .P.  D ) )
975, 14, 96syl2anc 403 . . . . . . . . . . . . . 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 5561 . . . . . . . . . . . . 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 2124 . . . . . . . . . . . 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 270 . . . . . . . . . . 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 6840 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 14  |-  ( ( ( B  +P.  C
)  e.  P.  /\  F  e.  P. )  ->  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) ) )
10461, 14, 103syl2anc 403 . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  =  ( F  .P.  B ) )
1061, 14, 105syl2anc 403 . . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  P.  /\  F  e.  P. )  ->  ( C  .P.  F
)  =  ( F  .P.  C ) )
10811, 14, 107syl2anc 403 . . . . . . . . . . . . . 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 5561 . . . . . . . . . . . . 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 2124 . . . . . . . . . . . 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 270 . . . . . . . . . . 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 2122 . . . . . . . . . 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 5558 . . . . . . . . 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 463 . . . . . . . 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 6824 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  F  e.  P. )  ->  ( C  .P.  F
)  e.  P. )
11611, 14, 115syl2anc 403 . . . . . . . . . . . 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 6824 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
11811, 26, 117syl2anc 403 . . . . . . . . . . . 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 6831 . . . . . . . . . . . 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 1170 . . . . . . . . . . 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 270 . . . . . . . . . 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 5551 . . . . . . . . . . . . 13  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
123122adantl 271 . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 2122 . . . . . . . . . . 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 5559 . . . . . . . . . 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 2114 . . . . . . . . 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 462 . . . . . . . 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 2114 . . . . . . 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 6824 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
13635, 14, 135syl2anc 403 . . . . . . . . 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 5712 . . . . . . . 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 270 . . . . . . 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 5713 . . . . . . . 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 270 . . . . . . 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 2122 . . . . . 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 5559 . . . . 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 2125 . . . 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 6789 . . . . . . 7  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( C  .P.  S )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  P. )
145136, 118, 144syl2anc 403 . . . . . 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 5715 . . . . 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 270 . . . 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 6789 . . . . . . 7  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( D  .P.  S ) )  e.  P. )
14937, 77, 148syl2anc 403 . . . . . 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 6831 . . . . . 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 1170 . . . . 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 270 . . . 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 2122 . . 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 6789 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
155154adantl 271 . . . . . 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 5716 . . . . 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 5559 . . . 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 270 . . 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 5718 . . . . 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 5559 . . . 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 270 . . 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 2122 . 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 113 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5543   P.cnp 6543    +P. cpp 6545    .P. cmp 6546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-imp 6721
This theorem is referenced by:  mulcmpblnr  6980
  Copyright terms: Public domain W3C validator