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Theorem mulcmpblnrlemg 7572
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 7404 . . . . . . . . 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 7404 . . . . . . . . 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 7369 . . . . . . . 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 7404 . . . . . . . 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 7404 . . . . . . . . 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 7404 . . . . . . . . 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 7369 . . . . . . . 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 7411 . . . . . . 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 1217 . . . . . 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 5790 . . . . . . . . . . 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 7420 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 7420 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 2181 . . . . . . . . 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 5798 . . . . . . . 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 7404 . . . . . . . . . . 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 7404 . . . . . . . . . . 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 7411 . . . . . . . . . 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 1217 . . . . . . . . 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 5789 . . . . . . . . . . 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 7420 . . . . . . . . . . . . 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 1217 . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . 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 2183 . . . . . . . . . . 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 7420 . . . . . . . . . . . . 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 1217 . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . 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 2183 . . . . . . . . . . 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 2181 . . . . . . . . 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 5797 . . . . . . . 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 2179 . . . . . . 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 7404 . . . . . . . . . 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 7404 . . . . . . . . . 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 7410 . . . . . . . . . 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 7411 . . . . . . . . . 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 5960 . . . . . . . 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 5959 . . . . . . . 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 2181 . . . . . 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 5797 . . . . 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 5789 . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . . 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 2183 . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . . 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 2183 . . . . . . . . . . . 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 2181 . . . . . . . . . 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 5797 . . . . . . . . 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 7404 . . . . . . . . . . . . 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 7404 . . . . . . . . . . . . 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 7411 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 5790 . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 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 2181 . . . . . . . . . . 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 5798 . . . . . . . . . 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 2173 . . . . . . . . 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 2173 . . . . . . 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 7404 . . . . . . . . . 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 5959 . . . . . . . 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 5960 . . . . . . . 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 2181 . . . . . 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 5798 . . . . 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 2184 . . . 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 7369 . . . . . . 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 5962 . . . . 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 7369 . . . . . . 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 7411 . . . . . 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 1217 . . . . 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 2181 . . 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 7369 . . . . . . 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 5963 . . . . 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 5798 . . . 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 5965 . . . . 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 5798 . . . 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 2181 . 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 1332   e. wcel 1481  (class class class)co 5782   P.cnp 7123   +P. cpp 7125   .P. cmp 7126
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-imp 7301
This theorem is referenced by:  mulcmpblnr  7573
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