ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemexbt Unicode version
Theorem caucvgprprlemexbt 7916
Description: Lemma for caucvgprpr 7922. 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 7915 . . . . . . 7 |- ( ph -> L e. P. )
7 caucvgprprlemexbt.q . . . . . . . 8 |- ( ph -> Q e. Q. )
8 nqprlu 7757 . . . . . . . 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 7747 . . . . . . 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 411 . . . . . 6 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
12 caucvgprprlemexbt.t . . . . . 6 |- ( ph -> T e. P. )
13 ltdfpr 7716 . . . . . 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 411 . . . . 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 147 . . . 4 |- ( ph -> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
166adantr 276 . . . . . . . 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 276 . . . . . . . 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 539 . . . . . . . 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 7830 . . . . . . 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 529 . . . . . . . . . 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 4090 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( p <Q u <-> p <Q y ) )
2221abbidv 2347 . . . . . . . . . . . . . . . 16 |- ( u = y -> { p | p <Q u } = { p | p <Q y } )
23 breq1 4089 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( u <Q q <-> y <Q q ) )
2423abbidv 2347 . . . . . . . . . . . . . . . 16 |- ( u = y -> { q | u <Q q } = { q | y <Q q } )
2522, 24opeq12d 3868 . . . . . . . . . . . . . . 15 |- ( u = y -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q y } , { q | y <Q q } >. )
2625breq2d 4098 . . . . . . . . . . . . . 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 2531 . . . . . . . . . . . . 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 5638 . . . . . . . . . . . . . 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 7573 . . . . . . . . . . . . . . . 16 |- Q. e. _V
3029rabex 4232 . . . . . . . . . . . . . . 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 4232 . . . . . . . . . . . . . . 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 6305 . . . . . . . . . . . . . 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 2250 . . . . . . . . . . . . 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 2963 . . . . . . . . . . . 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 120 . . . . . . . . . . 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 114 . . . . . . . . . 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 5635 . . . . . . . . . . . 12 |- ( r = b -> ( F ` r ) = ( F ` b ) )
39 opeq1 3860 . . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
4039eceq1d 6733 . . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
4140fveq2d 5639 . . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
4241breq2d 4098 . . . . . . . . . . . . . 14 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
4342abbidv 2347 . . . . . . . . . . . . 13 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
4441breq1d 4096 . . . . . . . . . . . . . 14 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
4544abbidv 2347 . . . . . . . . . . . . 13 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
4643, 45opeq12d 3868 . . . . . . . . . . . 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 6031 . . . . . . . . . . 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 4096 . . . . . . . . . 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 2766 . . . . . . . . 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 122 . . . . . . . 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 110 . . . . . . . . . . . . . . 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 7829 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
5352adantl 277 . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . . . . . 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, 55ffvelcdmd 5779 . . . . . . . . . . . . . . . . 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 7758 . . . . . . . . . . . . . . . . . 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 7747 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . 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 112 . . . . . . . . . . . . . . . . . 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 7757 . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . 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 7788 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
6867adantl 277 . . . . . . . . . . . . . . . 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 6187 . . . . . . . . . . . . . . 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 147 . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . 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 7771 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 4114 . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 4090 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( p <Q ( y +Q Q ) <-> p <Q x ) )
7877abbidv 2347 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { p | p <Q ( y +Q Q ) } = { p | p <Q x } )
79 breq1 4089 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( ( y +Q Q ) <Q q <-> x <Q q ) )
8079abbidv 2347 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { q | ( y +Q Q ) <Q q } = { q | x <Q q } )
8178, 80opeq12d 3868 . . . . . . . . . . . . . . 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 4098 . . . . . . . . . . . . . 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 147 . . . . . . . . . . . 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 535 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 7747 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 7762 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 167 . . . . . . . . . . 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 540 . . . . . . . . . . . 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 492 . . . . . . . . . . 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 306 . . . . . . . . . 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 115 . . . . . . . . 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 2632 . . . . . . . 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 2653 . . . . . 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 375 . . . . 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 2632 . . . 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 2695 . . 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 122 . 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 ) ) )
1042ffvelcdmda 5778 . . . . . 6 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
10557adantl 277 . . . . . 6 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
106104, 105, 59syl2anc 411 . . . . 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 276 . . . . 5 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
108106, 107, 87syl2anc 411 . . . 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 276 . . . 4 |- ( ( ph /\ b e. N. ) -> T e. P. )
110 ltdfpr 7716 . . . 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 411 . . 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 2527 . 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 167 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 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3670   class class class wbr 4086   -->wf 5320   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   1oc1o 6570   [cec 6695   N.cnpi 7482   <N clti 7485   ~Q ceq 7489   Q.cnq 7490   +Q cplq 7492   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   +P. cpp 7503   <P cltp 7505
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-iltp 7680
This theorem is referenced by:  caucvgprprlemexb  7917
  Copyright terms: Public domain W3C validator