ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemloc Unicode version
Theorem caucvgprprlemloc 8076
Description: Lemma for caucvgprpr 8085. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f |- ( ph -> F : N. --> P. )
caucvgprpr.cau |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprpr.bnd |- ( ph -> A. m e. N. A <P ( F ` m ) )
caucvgprpr.lim |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
Assertion
Ref Expression
caucvgprprlemloc |- ( ph -> A. s e. Q. A. t e. Q. ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
Distinct variable groups:   A, m   m, F   F, l, r   u, F, r   q, p, s, t   ph, s, t   p, l, q, s, t, r   u, p, q, s, t
Allowed substitution hints:   ph( u, k, m, n, r, q, p, l)   A( u, t, k, n, s, r, q, p, l)   F( t, k, n, s, q, p)   L( u, t, k, m, n, s, r, q, p, l)
Proof of Theorem caucvgprprlemloc
Dummy variables a b f g h c x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 7782 . . . . 5 |- ( s <Q t -> E. y e. Q. ( s +Q y ) = t )
21adantl 277 . . . 4 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> E. y e. Q. ( s +Q y ) = t )
3 subhalfnqq 7787 . . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
43ad2antrl 494 . . . . 5 |- ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) -> E. x e. Q. ( x +Q x ) <Q y )
5 archrecnq 8036 . . . . . . 7 |- ( x e. Q. -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
65ad2antrl 494 . . . . . 6 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
7 simpllr 540 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> s <Q t )
87adantr 276 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> s <Q t )
9 simplrl 541 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> y e. Q. )
109adantr 276 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> y e. Q. )
11 simplrr 542 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> ( s +Q y ) = t )
1211adantr 276 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q y ) = t )
13 simplrl 541 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> x e. Q. )
14 simplrr 542 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( x +Q x ) <Q y )
15 simprl 535 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> c e. N. )
16 simprr 537 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
178, 10, 12, 13, 14, 15, 16caucvgprprlemloccalc 8057 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
18 simplrl 541 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> s e. Q. )
1918ad3antrrr 496 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> s e. Q. )
20 nnnq 7795 . . . . . . . . . . . . . 14 |- ( c e. N. -> [ <. c , 1o >. ] ~Q e. Q. )
2120ad2antrl 494 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> [ <. c , 1o >. ] ~Q e. Q. )
22 recclnq 7765 . . . . . . . . . . . . 13 |- ( [ <. c , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
2321, 22syl 14 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
24 addclnq 7748 . . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
2519, 23, 24syl2anc 415 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
26 nqprlu 7920 . . . . . . . . . . 11 |- ( ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
2725, 26syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
28 nqprlu 7920 . . . . . . . . . . 11 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
2923, 28syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
30 addclpr 7910 . . . . . . . . . 10 |- ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. /\ <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
3127, 29, 30syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
32 simplrr 542 . . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> t e. Q. )
3332ad3antrrr 496 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> t e. Q. )
34 nqprlu 7920 . . . . . . . . . 10 |- ( t e. Q. -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
3533, 34syl 14 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
36 caucvgprpr.f . . . . . . . . . . . 12 |- ( ph -> F : N. --> P. )
3736ad5antr 500 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> F : N. --> P. )
3837, 15ffvelcdmd 5847 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( F ` c ) e. P. )
39 ltrelnq 7738 . . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
4039brel 4829 . . . . . . . . . . . . 13 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) )
4140simpld 112 . . . . . . . . . . . 12 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
4241ad2antll 495 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
4342, 28syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
44 addclpr 7910 . . . . . . . . . 10 |- ( ( ( F ` c ) e. P. /\ <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
4538, 43, 44syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
46 ltsopr 7969 . . . . . . . . . 10 |- <P Or P.
47 sowlin 4467 . . . . . . . . . 10 |- ( ( <P Or P. /\ ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q t } , { q | t <Q q } >. e. P. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
4846, 47mpan 428 . . . . . . . . 9 |- ( ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q t } , { q | t <Q q } >. e. P. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
4931, 35, 45, 48syl3anc 1278 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
5017, 49mpd 13 . . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
5119adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> s e. Q. )
52 simplrl 541 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> c e. N. )
53 simpr 110 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) )
54 ltaprg 7992 . . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
5554adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
5642adantr 276 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
5751, 56, 24syl2anc 415 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
5857, 26syl 14 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
5938adantr 276 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( F ` c ) e. P. )
6056, 28syl 14 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
61 addcomprg 7951 . . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
6261adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
6355, 58, 59, 60, 62caovord2d 6262 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) <-> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) )
6453, 63mpbird 167 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) )
65 opeq1 3904 . . . . . . . . . . . . . . . . . . 19 |- ( a = c -> <. a , 1o >. = <. c , 1o >. )
6665eceq1d 6847 . . . . . . . . . . . . . . . . . 18 |- ( a = c -> [ <. a , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
6766fveq2d 5702 . . . . . . . . . . . . . . . . 17 |- ( a = c -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
6867oveq2d 6104 . . . . . . . . . . . . . . . 16 |- ( a = c -> ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
6968breq2d 4143 . . . . . . . . . . . . . . 15 |- ( a = c -> ( p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
7069abbidv 2358 . . . . . . . . . . . . . 14 |- ( a = c -> { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } )
7168breq1d 4141 . . . . . . . . . . . . . . 15 |- ( a = c -> ( ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q ) )
7271abbidv 2358 . . . . . . . . . . . . . 14 |- ( a = c -> { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } = { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } )
7370, 72opeq12d 3912 . . . . . . . . . . . . 13 |- ( a = c -> <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. )
74 fveq2 5698 . . . . . . . . . . . . 13 |- ( a = c -> ( F ` a ) = ( F ` c ) )
7573, 74breq12d 4144 . . . . . . . . . . . 12 |- ( a = c -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) ) )
7675rspcev 2929 . . . . . . . . . . 11 |- ( ( c e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) ) -> E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) )
7752, 64, 76syl2anc 415 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) )
78 caucvgprpr.lim . . . . . . . . . . 11 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
7978caucvgprprlemell 8058 . . . . . . . . . 10 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) ) )
8051, 77, 79sylanbrc 421 . . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> s e. ( 1st ` L ) )
8180ex 115 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) -> s e. ( 1st ` L ) ) )
8233adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> t e. Q. )
83 fveq2 5698 . . . . . . . . . . . . . 14 |- ( b = c -> ( F ` b ) = ( F ` c ) )
84 opeq1 3904 . . . . . . . . . . . . . . . . . . 19 |- ( b = c -> <. b , 1o >. = <. c , 1o >. )
8584eceq1d 6847 . . . . . . . . . . . . . . . . . 18 |- ( b = c -> [ <. b , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
8685fveq2d 5702 . . . . . . . . . . . . . . . . 17 |- ( b = c -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
8786breq2d 4143 . . . . . . . . . . . . . . . 16 |- ( b = c -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
8887abbidv 2358 . . . . . . . . . . . . . . 15 |- ( b = c -> { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } )
8986breq1d 4141 . . . . . . . . . . . . . . . 16 |- ( b = c -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q ) )
9089abbidv 2358 . . . . . . . . . . . . . . 15 |- ( b = c -> { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } )
9188, 90opeq12d 3912 . . . . . . . . . . . . . 14 |- ( b = c -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. )
9283, 91oveq12d 6106 . . . . . . . . . . . . 13 |- ( b = c -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) )
9392breq1d 4141 . . . . . . . . . . . 12 |- ( b = c -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. <-> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
9493rspcev 2929 . . . . . . . . . . 11 |- ( ( c e. N. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
9515, 94sylan 283 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
9678caucvgprprlemelu 8059 . . . . . . . . . 10 |- ( t e. ( 2nd ` L ) <-> ( t e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
9782, 95, 96sylanbrc 421 . . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> t e. ( 2nd ` L ) )
9897ex 115 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> t e. ( 2nd ` L ) ) )
9981, 98orim12d 798 . . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
10050, 99mpd 13 . . . . . 6 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
1016, 100rexlimddv 2673 . . . . 5 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
1024, 101rexlimddv 2673 . . . 4 |- ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
1032, 102rexlimddv 2673 . . 3 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
104103ex 115 . 2 |- ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) -> ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
105104ralrimivva 2632 1 |- ( ph -> A. s e. Q. A. t e. Q. ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   Or wor 4442   -->wf 5376   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1oc1o 6684   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   +Q cplq 7655   *Qcrq 7657   <Q cltq 7658   P.cnp 7664   +P. cpp 7666   <P cltp 7668
This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737
This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843
This theorem is used by:  caucvgprprlemcl  8077
  Copyright terms: Public domain W3C validator