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Theorem mulcmpblnrlemg 7702
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 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. ) ) ) -> B e. P. )
2 simprlr 533 . . . . . . . . 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 7534 . . . . . . . . 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 531 . . . . . . . . 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 534 . . . . . . . . 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 7534 . . . . . . . . 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 7499 . . . . . . . 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 530 . . . . . . . 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 7534 . . . . . . . 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 532 . . . . . . . . 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 7534 . . . . . . . . 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 7534 . . . . . . . . 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 7499 . . . . . . . 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 7541 . . . . . . 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 1233 . . . . . 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 5861 . . . . . . . . . . 11 |- ( ( F +P. S ) = ( G +P. R ) -> ( D .P. ( F +P. S ) ) = ( D .P. ( G +P. R ) ) )
2524ad2antll 488 . . . . . . . . . 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 535 . . . . . . . . . . . 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 7550 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 7550 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 2211 . . . . . . . . 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 5869 . . . . . . . 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 528 . . . . . . . . . . 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 7534 . . . . . . . . . . 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 7534 . . . . . . . . . . 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 7541 . . . . . . . . . 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 1233 . . . . . . . . 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 5860 . . . . . . . . . . 11 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A +P. D ) .P. G ) = ( ( B +P. C ) .P. G ) )
4443ad2antrl 487 . . . . . . . . . 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 7550 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . 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 7499 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . 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 2213 . . . . . . . . . . 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 7550 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . 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 7499 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . 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 2213 . . . . . . . . . . 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 2211 . . . . . . . . 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 5868 . . . . . . . 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 2209 . . . . . . 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 7534 . . . . . . . . . 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 7534 . . . . . . . . . 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 7540 . . . . . . . . . 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 7541 . . . . . . . . . 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 6034 . . . . . . . 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 6033 . . . . . . . 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 2211 . . . . . 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 5868 . . . . 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 5860 . . . . . . . . . . . 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 7550 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . . 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 2213 . . . . . . . . . . . 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 7550 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . . 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 7542 . . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . . 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 2213 . . . . . . . . . . . 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 2211 . . . . . . . . . 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 5868 . . . . . . . . 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 476 . . . . . . . 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 7534 . . . . . . . . . . . . 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 7534 . . . . . . . . . . . . 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 7541 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 5861 . . . . . . . . . . . . 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 7550 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 7550 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 2211 . . . . . . . . . . 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 5869 . . . . . . . . . 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 2203 . . . . . . . . 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 475 . . . . . . . 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 2203 . . . . . . 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 7534 . . . . . . . . . 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 6033 . . . . . . . 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 6034 . . . . . . . 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 2211 . . . . . 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 5869 . . . . 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 2214 . . . 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 7499 . . . . . . 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 6036 . . . . 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 7499 . . . . . . 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 7541 . . . . . 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 1233 . . . . 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 2211 . . 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 7499 . . . . . . 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 6037 . . . . 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 5869 . . . 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 6039 . . . . 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 5869 . . . 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 2211 . 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 973   = wceq 1348   e. wcel 2141  (class class class)co 5853   P.cnp 7253   +P. cpp 7255   .P. cmp 7256
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431
This theorem is referenced by:  mulcmpblnr  7703
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