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Theorem mulcmpblnrlemg 7852
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 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. ) ) ) -> B e. P. )
2 simprlr 538 . . . . . . . . 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 7684 . . . . . . . . 9 |- ( ( B e. P. /\ G e. P. ) -> ( B .P. G ) e. P. )
41, 2, 3syl2anc 411 . . . . . . . 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 536 . . . . . . . . 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 539 . . . . . . . . 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 7684 . . . . . . . . 9 |- ( ( D e. P. /\ R e. P. ) -> ( D .P. R ) e. P. )
85, 6, 7syl2anc 411 . . . . . . . 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 7649 . . . . . . . 8 |- ( ( ( B .P. G ) e. P. /\ ( D .P. R ) e. P. ) -> ( ( B .P. G ) +P. ( D .P. R ) ) e. P. )
104, 8, 9syl2anc 411 . . . . . . 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 535 . . . . . . . 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 7684 . . . . . . . 8 |- ( ( C e. P. /\ G e. P. ) -> ( C .P. G ) e. P. )
1311, 2, 12syl2anc 411 . . . . . . 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 537 . . . . . . . . 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 7684 . . . . . . . . 9 |- ( ( B e. P. /\ F e. P. ) -> ( B .P. F ) e. P. )
161, 14, 15syl2anc 411 . . . . . . . 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 7684 . . . . . . . . 9 |- ( ( C e. P. /\ R e. P. ) -> ( C .P. R ) e. P. )
1811, 6, 17syl2anc 411 . . . . . . . 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 7649 . . . . . . . 8 |- ( ( ( B .P. F ) e. P. /\ ( C .P. R ) e. P. ) -> ( ( B .P. F ) +P. ( C .P. R ) ) e. P. )
2016, 18, 19syl2anc 411 . . . . . . 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 7691 . . . . . . 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 1249 . . . . . 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 276 . . . . 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 5951 . . . . . . . . . . 11 |- ( ( F +P. S ) = ( G +P. R ) -> ( D .P. ( F +P. S ) ) = ( D .P. ( G +P. R ) ) )
2524ad2antll 491 . . . . . . . . . 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 540 . . . . . . . . . . . 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 7700 . . . . . . . . . . . 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 1249 . . . . . . . . . . 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 276 . . . . . . . . . 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 7700 . . . . . . . . . . . 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 1249 . . . . . . . . . . 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 276 . . . . . . . . . 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 2245 . . . . . . . . 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 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. 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 533 . . . . . . . . . . 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 7684 . . . . . . . . . . 11 |- ( ( A e. P. /\ G e. P. ) -> ( A .P. G ) e. P. )
3735, 2, 36syl2anc 411 . . . . . . . . . 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 7684 . . . . . . . . . . 11 |- ( ( D e. P. /\ G e. P. ) -> ( D .P. G ) e. P. )
395, 2, 38syl2anc 411 . . . . . . . . . 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 7691 . . . . . . . . . 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 1249 . . . . . . . . 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 276 . . . . . . . 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 5950 . . . . . . . . . . 11 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A +P. D ) .P. G ) = ( ( B +P. C ) .P. G ) )
4443ad2antrl 490 . . . . . . . . . 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 7700 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 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 7649 . . . . . . . . . . . . . 14 |- ( ( A e. P. /\ D e. P. ) -> ( A +P. D ) e. P. )
4835, 5, 47syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . 13 |- ( ( ( A +P. D ) e. P. /\ G e. P. ) -> ( ( A +P. D ) .P. G ) = ( G .P. ( A +P. D ) ) )
5048, 2, 49syl2anc 411 . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( A e. P. /\ G e. P. ) -> ( A .P. G ) = ( G .P. A ) )
5235, 2, 51syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( D e. P. /\ G e. P. ) -> ( D .P. G ) = ( G .P. D ) )
545, 2, 53syl2anc 411 . . . . . . . . . . . . 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 5961 . . . . . . . . . . . 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 2247 . . . . . . . . . . 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 276 . . . . . . . . . 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 7700 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 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 7649 . . . . . . . . . . . . . 14 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
611, 11, 60syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . 13 |- ( ( ( B +P. C ) e. P. /\ G e. P. ) -> ( ( B +P. C ) .P. G ) = ( G .P. ( B +P. C ) ) )
6361, 2, 62syl2anc 411 . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( B e. P. /\ G e. P. ) -> ( B .P. G ) = ( G .P. B ) )
651, 2, 64syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( C e. P. /\ G e. P. ) -> ( C .P. G ) = ( G .P. C ) )
6711, 2, 66syl2anc 411 . . . . . . . . . . . . 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 5961 . . . . . . . . . . . 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 2247 . . . . . . . . . . 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 276 . . . . . . . . . 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 2245 . . . . . . . . 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 5958 . . . . . . . 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 2243 . . . . . . 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 7684 . . . . . . . . . 10 |- ( ( D e. P. /\ F e. P. ) -> ( D .P. F ) e. P. )
755, 14, 74syl2anc 411 . . . . . . . . 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 7684 . . . . . . . . . 10 |- ( ( D e. P. /\ S e. P. ) -> ( D .P. S ) e. P. )
775, 26, 76syl2anc 411 . . . . . . . . 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 7690 . . . . . . . . . 10 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) = ( y +P. x ) )
7978adantl 277 . . . . . . . . 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 7691 . . . . . . . . . 10 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( ( x +P. y ) +P. z ) = ( x +P. ( y +P. z ) ) )
8180adantl 277 . . . . . . . . 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 6127 . . . . . . . 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 276 . . . . . . 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 6126 . . . . . . . 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 276 . . . . . . 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 2245 . . . . . 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 5958 . . . . 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 5950 . . . . . . . . . . . 12 |- ( ( A +P. D ) = ( B +P. C ) -> ( ( A +P. D ) .P. F ) = ( ( B +P. C ) .P. F ) )
8988adantl 277 . . . . . . . . . . 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 7700 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( ( A +P. D ) e. P. /\ F e. P. ) -> ( ( A +P. D ) .P. F ) = ( F .P. ( A +P. D ) ) )
9348, 14, 92syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ F e. P. ) -> ( A .P. F ) = ( F .P. A ) )
9535, 14, 94syl2anc 411 . . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . . 15 |- ( ( D e. P. /\ F e. P. ) -> ( D .P. F ) = ( F .P. D ) )
975, 14, 96syl2anc 411 . . . . . . . . . . . . . 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 5961 . . . . . . . . . . . . 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 2247 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 7700 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . 14 |- ( ( ( B +P. C ) e. P. /\ F e. P. ) -> ( ( B +P. C ) .P. F ) = ( F .P. ( B +P. C ) ) )
10461, 14, 103syl2anc 411 . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ F e. P. ) -> ( B .P. F ) = ( F .P. B ) )
1061, 14, 105syl2anc 411 . . . . . . . . . . . . . 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 7692 . . . . . . . . . . . . . . 15 |- ( ( C e. P. /\ F e. P. ) -> ( C .P. F ) = ( F .P. C ) )
10811, 14, 107syl2anc 411 . . . . . . . . . . . . . 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 5961 . . . . . . . . . . . . 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 2247 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 2245 . . . . . . . . . 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 5958 . . . . . . . . 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 479 . . . . . . . 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 7684 . . . . . . . . . . . . 13 |- ( ( C e. P. /\ F e. P. ) -> ( C .P. F ) e. P. )
11611, 14, 115syl2anc 411 . . . . . . . . . . . 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 7684 . . . . . . . . . . . . 13 |- ( ( C e. P. /\ S e. P. ) -> ( C .P. S ) e. P. )
11811, 26, 117syl2anc 411 . . . . . . . . . . . 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 7691 . . . . . . . . . . . 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 1249 . . . . . . . . . . 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 276 . . . . . . . . . 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 5951 . . . . . . . . . . . . 13 |- ( ( F +P. S ) = ( G +P. R ) -> ( C .P. ( F +P. S ) ) = ( C .P. ( G +P. R ) ) )
123122adantl 277 . . . . . . . . . . . 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 7700 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 7700 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 2245 . . . . . . . . . . 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 5959 . . . . . . . . . 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 2237 . . . . . . . . 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 478 . . . . . . . 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 2237 . . . . . . 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 7684 . . . . . . . . . 10 |- ( ( A e. P. /\ F e. P. ) -> ( A .P. F ) e. P. )
13635, 14, 135syl2anc 411 . . . . . . . . 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 6126 . . . . . . . 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 276 . . . . . . 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 6127 . . . . . . . 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 276 . . . . . . 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 2245 . . . . . 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 5959 . . . . 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 2248 . . . 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 7649 . . . . . . 7 |- ( ( ( A .P. F ) e. P. /\ ( C .P. S ) e. P. ) -> ( ( A .P. F ) +P. ( C .P. S ) ) e. P. )
145136, 118, 144syl2anc 411 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( A .P. F ) +P. ( C .P. S ) ) e. P. )
14610, 145, 75, 79, 81caov13d 6129 . . . . 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 276 . . . 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 7649 . . . . . . 7 |- ( ( ( A .P. G ) e. P. /\ ( D .P. S ) e. P. ) -> ( ( A .P. G ) +P. ( D .P. S ) ) e. P. )
14937, 77, 148syl2anc 411 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( A .P. G ) +P. ( D .P. S ) ) e. P. )
150 addassprg 7691 . . . . . 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 1249 . . . . 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 276 . . . 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 2245 . . 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 7649 . . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) e. P. )
155154adantl 277 . . . . . 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 6130 . . . . 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 5959 . . . 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 276 . . 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 6132 . . . . 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 5959 . . . 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 276 . . 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 2245 . 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 115 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 104   /\ w3a 980   = wceq 1372   e. wcel 2175  (class class class)co 5943   P.cnp 7403   +P. cpp 7405   .P. cmp 7406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iplp 7580  df-imp 7581
This theorem is referenced by:  mulcmpblnr  7853
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