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Theorem distrlem4pru 7942
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4pru |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
Distinct variable groups:   x, y, z, f, A   x, B, y, z, f   x, C, y, z, f
Proof of Theorem distrlem4pru
Dummy variables w v u g h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 7758 . . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
21adantl 277 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u .Q w ) <Q ( u .Q v ) ) )
3 simp1 1028 . . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> A e. P. )
4 simpll 531 . . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> x e. ( 2nd ` A ) )
5 prop 7832 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6 elprnqu 7839 . . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
75, 6sylan 283 . . . . . . 7 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
83, 4, 7syl2an 289 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> x e. Q. )
9 simprl 535 . . . . . . 7 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> f e. ( 2nd ` A ) )
10 elprnqu 7839 . . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
115, 10sylan 283 . . . . . . 7 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
123, 9, 11syl2an 289 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> f e. Q. )
13 simpl3 1033 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> C e. P. )
14 simprrr 546 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> z e. ( 2nd ` C ) )
15 prop 7832 . . . . . . . 8 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
16 elprnqu 7839 . . . . . . . 8 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
1715, 16sylan 283 . . . . . . 7 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
1813, 14, 17syl2anc 415 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> z e. Q. )
19 mulcomnqg 7740 . . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w .Q v ) = ( v .Q w ) )
2019adantl 277 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w .Q v ) = ( v .Q w ) )
212, 8, 12, 18, 20caovord2d 6249 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f <-> ( x .Q z ) <Q ( f .Q z ) ) )
22 mulclnq 7733 . . . . . . 7 |- ( ( x e. Q. /\ z e. Q. ) -> ( x .Q z ) e. Q. )
238, 18, 22syl2anc 415 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q z ) e. Q. )
24 mulclnq 7733 . . . . . . 7 |- ( ( f e. Q. /\ z e. Q. ) -> ( f .Q z ) e. Q. )
2512, 18, 24syl2anc 415 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q z ) e. Q. )
26 simpl2 1032 . . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> B e. P. )
27 simprlr 544 . . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> y e. ( 2nd ` B ) )
28 prop 7832 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
29 elprnqu 7839 . . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
3028, 29sylan 283 . . . . . . . 8 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
3126, 27, 30syl2anc 415 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> y e. Q. )
32 mulclnq 7733 . . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) e. Q. )
338, 31, 32syl2anc 415 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q y ) e. Q. )
34 ltanqg 7757 . . . . . 6 |- ( ( ( x .Q z ) e. Q. /\ ( f .Q z ) e. Q. /\ ( x .Q y ) e. Q. ) -> ( ( x .Q z ) <Q ( f .Q z ) <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
3523, 25, 33, 34syl3anc 1278 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q z ) <Q ( f .Q z ) <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
3621, 35bitrd 188 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f <-> ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
37 simpl1 1031 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> A e. P. )
38 addclpr 7894 . . . . . . . 8 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
39383adant1 1046 . . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
4039adantr 276 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( B +P. C ) e. P. )
41 mulclpr 7929 . . . . . 6 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( A .P. ( B +P. C ) ) e. P. )
4237, 40, 41syl2anc 415 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( A .P. ( B +P. C ) ) e. P. )
43 distrnqg 7744 . . . . . . 7 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
448, 31, 18, 43syl3anc 1278 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
45 simprll 543 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> x e. ( 2nd ` A ) )
46 df-iplp 7825 . . . . . . . . . 10 |- +P. = ( u e. P. , v e. P. |-> <. { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` u ) /\ h e. ( 1st ` v ) /\ w = ( g +Q h ) ) } , { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` u ) /\ h e. ( 2nd ` v ) /\ w = ( g +Q h ) ) } >. )
47 addclnq 7732 . . . . . . . . . 10 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
4846, 47genppreclu 7872 . . . . . . . . 9 |- ( ( B e. P. /\ C e. P. ) -> ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) )
4948imp 124 . . . . . . . 8 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) )
5026, 13, 27, 14, 49syl22anc 1279 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) )
51 df-imp 7826 . . . . . . . . 9 |- .P. = ( u e. P. , v e. P. |-> <. { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` u ) /\ h e. ( 1st ` v ) /\ w = ( g .Q h ) ) } , { w e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` u ) /\ h e. ( 2nd ` v ) /\ w = ( g .Q h ) ) } >. )
52 mulclnq 7733 . . . . . . . . 9 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
5351, 52genppreclu 7872 . . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( x e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
5453imp 124 . . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( x e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
5537, 40, 45, 50, 54syl22anc 1279 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
5644, 55eqeltrrd 2316 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
57 prop 7832 . . . . . 6 |- ( ( A .P. ( B +P. C ) ) e. P. -> <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. )
58 prcunqu 7842 . . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
5957, 58sylan 283 . . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
6042, 56, 59syl2anc 415 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( ( x .Q y ) +Q ( x .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
6136, 60sylbid 150 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x <Q f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
622, 12, 8, 31, 20caovord2d 6249 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x <-> ( f .Q y ) <Q ( x .Q y ) ) )
63 ltanqg 7757 . . . . . . 7 |- ( ( w e. Q. /\ v e. Q. /\ u e. Q. ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
6463adantl 277 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. /\ u e. Q. ) ) -> ( w <Q v <-> ( u +Q w ) <Q ( u +Q v ) ) )
65 mulclnq 7733 . . . . . . 7 |- ( ( f e. Q. /\ y e. Q. ) -> ( f .Q y ) e. Q. )
6612, 31, 65syl2anc 415 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q y ) e. Q. )
67 addcomnqg 7738 . . . . . . 7 |- ( ( w e. Q. /\ v e. Q. ) -> ( w +Q v ) = ( v +Q w ) )
6867adantl 277 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ ( w e. Q. /\ v e. Q. ) ) -> ( w +Q v ) = ( v +Q w ) )
6964, 66, 33, 25, 68caovord2d 6249 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( f .Q y ) <Q ( x .Q y ) <-> ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
7062, 69bitrd 188 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x <-> ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) ) )
71 distrnqg 7744 . . . . . . 7 |- ( ( f e. Q. /\ y e. Q. /\ z e. Q. ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
7212, 31, 18, 71syl3anc 1278 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) = ( ( f .Q y ) +Q ( f .Q z ) ) )
73 simprrl 545 . . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> f e. ( 2nd ` A ) )
7451, 52genppreclu 7872 . . . . . . . 8 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( ( f e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
7574imp 124 . . . . . . 7 |- ( ( ( A e. P. /\ ( B +P. C ) e. P. ) /\ ( f e. ( 2nd ` A ) /\ ( y +Q z ) e. ( 2nd ` ( B +P. C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
7637, 40, 73, 50, 75syl22anc 1279 . . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
7772, 76eqeltrrd 2316 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
78 prcunqu 7842 . . . . . 6 |- ( ( <. ( 1st ` ( A .P. ( B +P. C ) ) ) , ( 2nd ` ( A .P. ( B +P. C ) ) ) >. e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
7957, 78sylan 283 . . . . 5 |- ( ( ( A .P. ( B +P. C ) ) e. P. /\ ( ( f .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
8042, 77, 79syl2anc 415 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( ( f .Q y ) +Q ( f .Q z ) ) <Q ( ( x .Q y ) +Q ( f .Q z ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
8170, 80sylbid 150 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( f <Q x -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
8261, 81jaod 729 . 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
83 ltsonq 7755 . . . . 5 |- <Q Or Q.
84 nqtri3or 7753 . . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f \/ x = f \/ f <Q x ) )
8583, 84sotritrieq 4465 . . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
868, 12, 85syl2anc 415 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x = f <-> -. ( x <Q f \/ f <Q x ) ) )
87 oveq1 6082 . . . . . . 7 |- ( x = f -> ( x .Q z ) = ( f .Q z ) )
8887oveq2d 6091 . . . . . 6 |- ( x = f -> ( ( x .Q y ) +Q ( x .Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
8944, 88sylan9eq 2291 . . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
9055adantr 276 . . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( x .Q ( y +Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
9189, 90eqeltrrd 2316 . . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) /\ x = f ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
9291ex 115 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( x = f -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
9386, 92sylbird 170 . 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( -. ( x <Q f \/ f <Q x ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
94 ltdcnq 7754 . . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID x <Q f )
95 ltdcnq 7754 . . . . . 6 |- ( ( f e. Q. /\ x e. Q. ) -> DECID f <Q x )
9695ancoms 268 . . . . 5 |- ( ( x e. Q. /\ f e. Q. ) -> DECID f <Q x )
97 dcor 948 . . . . 5 |- (DECID x <Q f -> (DECID f <Q x -> DECID ( x <Q f \/ f <Q x ) ) )
9894, 96, 97sylc 62 . . . 4 |- ( ( x e. Q. /\ f e. Q. ) -> DECID ( x <Q f \/ f <Q x ) )
998, 12, 98syl2anc 415 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> DECID ( x <Q f \/ f <Q x ) )
100 df-dc 847 . . 3 |- (DECID ( x <Q f \/ f <Q x ) <-> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
10199, 100sylib 122 . 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x <Q f \/ f <Q x ) \/ -. ( x <Q f \/ f <Q x ) ) )
10282, 93, 101mpjaod 730 1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   +Q cplq 7639   .Q cmq 7640   <Q cltq 7642   P.cnp 7648   +P. cpp 7650   .P. cmp 7651
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-imp 7826
This theorem is referenced by:  distrlem5pru  7944
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