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Theorem caucvgprprlemexbt 6896
Description: Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
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 } >. } >.
caucvgprprlemexbt.q |- ( ph -> Q e. Q. )
caucvgprprlemexbt.t |- ( ph -> T e. P. )
caucvgprprlemexbt.lt |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )
Assertion
Ref Expression
caucvgprprlemexbt |- ( ph -> 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 Q } , { q | Q <Q q } >. ) <P T )
Distinct variable groups:   A, m   m, F   A, r, m   F, b   k, F, l, n, u   F, r   L, b   k, L   Q, b, p, q   T, b   ph, b   r, b, p, q   k, p, q, r, l, u
Allowed substitution hints:   ph( u, k, m, n, r, q, p, l)   A( u, k, n, q, p, b, l)   Q( u, k, m, n, r, l)   T( u, k, m, n, r, q, p, l)   F( q, p)   L( u, m, n, r, q, p, l)
Proof of Theorem caucvgprprlemexbt
Dummy variables f g h x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemexbt.lt . . . . 5 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )
2 caucvgprpr.f . . . . . . . 8 |- ( ph -> F : N. --> P. )
3 caucvgprpr.cau . . . . . . . 8 |- ( 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 } >. ) ) ) )
4 caucvgprpr.bnd . . . . . . . 8 |- ( ph -> A. m e. N. A <P ( F ` m ) )
5 caucvgprpr.lim . . . . . . . 8 |- 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 } >. } >.
62, 3, 4, 5caucvgprprlemclphr 6895 . . . . . . 7 |- ( ph -> L e. P. )
7 caucvgprprlemexbt.q . . . . . . . 8 |- ( ph -> Q e. Q. )
8 nqprlu 6737 . . . . . . . 8 |- ( Q e. Q. -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
97, 8syl 14 . . . . . . 7 |- ( ph -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
10 addclpr 6727 . . . . . . 7 |- ( ( L e. P. /\ <. { p | p <Q Q } , { q | Q <Q q } >. e. P. ) -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
116, 9, 10syl2anc 403 . . . . . 6 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
12 caucvgprprlemexbt.t . . . . . 6 |- ( ph -> T e. P. )
13 ltdfpr 6696 . . . . . 6 |- ( ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. /\ T e. P. ) -> ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
1411, 12, 13syl2anc 403 . . . . 5 |- ( ph -> ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
151, 14mpbid 145 . . . 4 |- ( ph -> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
166adantr 270 . . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> L e. P. )
177adantr 270 . . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> Q e. Q. )
18 simprrl 505 . . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
1916, 17, 18prplnqu 6810 . . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> E. y e. ( 2nd ` L ) ( y +Q Q ) = x )
20 simprl 497 . . . . . . . . . 10 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> y e. ( 2nd ` L ) )
21 breq2 3789 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( p <Q u <-> p <Q y ) )
2221abbidv 2196 . . . . . . . . . . . . . . . 16 |- ( u = y -> { p | p <Q u } = { p | p <Q y } )
23 breq1 3788 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( u <Q q <-> y <Q q ) )
2423abbidv 2196 . . . . . . . . . . . . . . . 16 |- ( u = y -> { q | u <Q q } = { q | y <Q q } )
2522, 24opeq12d 3578 . . . . . . . . . . . . . . 15 |- ( u = y -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q y } , { q | y <Q q } >. )
2625breq2d 3797 . . . . . . . . . . . . . 14 |- ( u = y -> ( ( ( 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 } >. <-> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
2726rexbidv 2369 . . . . . . . . . . . . 13 |- ( u = y -> ( 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 } >. <-> 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 y } , { q | y <Q q } >. ) )
285fveq2i 5201 . . . . . . . . . . . . . 14 |- ( 2nd ` L ) = ( 2nd ` <. { 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 } >. } >. )
29 nqex 6553 . . . . . . . . . . . . . . . 16 |- Q. e. _V
3029rabex 3922 . . . . . . . . . . . . . . 15 |- { 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 ) } e. _V
3129rabex 3922 . . . . . . . . . . . . . . 15 |- { 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 } >. } e. _V
3230, 31op2nd 5794 . . . . . . . . . . . . . 14 |- ( 2nd ` <. { 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 } >. } >. ) = { 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 } >. }
3328, 32eqtri 2101 . . . . . . . . . . . . 13 |- ( 2nd ` L ) = { 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 } >. }
3427, 33elrab2 2751 . . . . . . . . . . . 12 |- ( y e. ( 2nd ` L ) <-> ( y 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 y } , { q | y <Q q } >. ) )
3534biimpi 118 . . . . . . . . . . 11 |- ( y e. ( 2nd ` L ) -> ( y 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 y } , { q | y <Q q } >. ) )
3635simprd 112 . . . . . . . . . 10 |- ( y e. ( 2nd ` L ) -> 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 y } , { q | y <Q q } >. )
3720, 36syl 14 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> 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 y } , { q | y <Q q } >. )
38 fveq2 5198 . . . . . . . . . . . 12 |- ( r = b -> ( F ` r ) = ( F ` b ) )
39 opeq1 3570 . . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
4039eceq1d 6165 . . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
4140fveq2d 5202 . . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
4241breq2d 3797 . . . . . . . . . . . . . 14 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
4342abbidv 2196 . . . . . . . . . . . . 13 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
4441breq1d 3795 . . . . . . . . . . . . . 14 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
4544abbidv 2196 . . . . . . . . . . . . 13 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
4643, 45opeq12d 3578 . . . . . . . . . . . 12 |- ( r = b -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. )
4738, 46oveq12d 5550 . . . . . . . . . . 11 |- ( r = b -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) )
4847breq1d 3795 . . . . . . . . . 10 |- ( r = b -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
4948cbvrexv 2578 . . . . . . . . 9 |- ( 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 y } , { q | y <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 y } , { q | y <Q q } >. )
5037, 49sylib 120 . . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> 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 y } , { q | y <Q q } >. )
51 simpr 108 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
52 ltaprg 6809 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
5352adantl 271 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
542ad4antr 477 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> F : N. --> P. )
55 simplr 496 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> b e. N. )
5654, 55ffvelrnd 5324 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( F ` b ) e. P. )
57 recnnpr 6738 . . . . . . . . . . . . . . . . . 18 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
5855, 57syl 14 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
59 addclpr 6727 . . . . . . . . . . . . . . . . 17 |- ( ( ( F ` b ) e. P. /\ <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
6056, 58, 59syl2anc 403 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
6120ad2antrr 471 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> y e. ( 2nd ` L ) )
6235simpld 110 . . . . . . . . . . . . . . . . . 18 |- ( y e. ( 2nd ` L ) -> y e. Q. )
6361, 62syl 14 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> y e. Q. )
64 nqprlu 6737 . . . . . . . . . . . . . . . . 17 |- ( y e. Q. -> <. { p | p <Q y } , { q | y <Q q } >. e. P. )
6563, 64syl 14 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q y } , { q | y <Q q } >. e. P. )
669ad4antr 477 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
67 addcomprg 6768 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
6867adantl 271 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
6953, 60, 65, 66, 68caovord2d 5690 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
7051, 69mpbid 145 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
717ad4antr 477 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> Q e. Q. )
72 addnqpr 6751 . . . . . . . . . . . . . . 15 |- ( ( y e. Q. /\ Q e. Q. ) -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
7363, 71, 72syl2anc 403 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
7470, 73breqtrrd 3811 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. )
75 simplrr 502 . . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) -> ( y +Q Q ) = x )
7675adantr 270 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( y +Q Q ) = x )
77 breq2 3789 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( p <Q ( y +Q Q ) <-> p <Q x ) )
7877abbidv 2196 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { p | p <Q ( y +Q Q ) } = { p | p <Q x } )
79 breq1 3788 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( ( y +Q Q ) <Q q <-> x <Q q ) )
8079abbidv 2196 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { q | ( y +Q Q ) <Q q } = { q | x <Q q } )
8178, 80opeq12d 3578 . . . . . . . . . . . . . . 15 |- ( ( y +Q Q ) = x -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = <. { p | p <Q x } , { q | x <Q q } >. )
8281breq2d 3797 . . . . . . . . . . . . . 14 |- ( ( y +Q Q ) = x -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
8376, 82syl 14 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
8474, 83mpbid 145 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. )
85 simplrl 501 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> x e. Q. )
8685ad2antrr 471 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. Q. )
87 addclpr 6727 . . . . . . . . . . . . . 14 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q Q } , { q | Q <Q q } >. e. P. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
8860, 66, 87syl2anc 403 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
89 nqpru 6742 . . . . . . . . . . . . 13 |- ( ( x e. Q. /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
9086, 88, 89syl2anc 403 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
9184, 90mpbird 165 . . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
92 simprrr 506 . . . . . . . . . . . 12 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> x e. ( 1st ` T ) )
9392ad3antrrr 475 . . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. ( 1st ` T ) )
9491, 93jca 300 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
9594ex 113 . . . . . . . . 9 |- ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
9695reximdva 2463 . . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> ( 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 y } , { q | y <Q q } >. -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
9750, 96mpd 13 . . . . . . 7 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
9819, 97rexlimddv 2481 . . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
9998expr 367 . . . . 5 |- ( ( ph /\ x e. Q. ) -> ( ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
10099reximdva 2463 . . . 4 |- ( ph -> ( E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) -> E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
10115, 100mpd 13 . . 3 |- ( ph -> E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
102 rexcom 2518 . . 3 |- ( E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) <-> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
103101, 102sylib 120 . 2 |- ( ph -> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
1042ffvelrnda 5323 . . . . . 6 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
10557adantl 271 . . . . . 6 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
106104, 105, 59syl2anc 403 . . . . 5 |- ( ( ph /\ b e. N. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
1079adantr 270 . . . . 5 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
108106, 107, 87syl2anc 403 . . . 4 |- ( ( ph /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
10912adantr 270 . . . 4 |- ( ( ph /\ b e. N. ) -> T e. P. )
110 ltdfpr 6696 . . . 4 |- ( ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. /\ T e. P. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
111108, 109, 110syl2anc 403 . . 3 |- ( ( ph /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
112111rexbidva 2365 . 2 |- ( ph -> ( 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 Q } , { q | Q <Q q } >. ) <P T <-> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
113103, 112mpbird 165 1 |- ( ph -> 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 Q } , { q | Q <Q q } >. ) <P T )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   +P. cpp 6483   <P cltp 6485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660
This theorem is referenced by:  caucvgprprlemexb  6897
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