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Theorem caucvgprprlemexbt 7507
Description: Lemma for caucvgprpr 7513. 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 7506 . . . . . . 7 |- ( ph -> L e. P. )
7 caucvgprprlemexbt.q . . . . . . . 8 |- ( ph -> Q e. Q. )
8 nqprlu 7348 . . . . . . . 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 7338 . . . . . . 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 408 . . . . . 6 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
12 caucvgprprlemexbt.t . . . . . 6 |- ( ph -> T e. P. )
13 ltdfpr 7307 . . . . . 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 408 . . . . 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 146 . . . 4 |- ( ph -> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
166adantr 274 . . . . . . . 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 274 . . . . . . . 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 528 . . . . . . . 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 7421 . . . . . . 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 520 . . . . . . . . . 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 3928 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( p <Q u <-> p <Q y ) )
2221abbidv 2255 . . . . . . . . . . . . . . . 16 |- ( u = y -> { p | p <Q u } = { p | p <Q y } )
23 breq1 3927 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( u <Q q <-> y <Q q ) )
2423abbidv 2255 . . . . . . . . . . . . . . . 16 |- ( u = y -> { q | u <Q q } = { q | y <Q q } )
2522, 24opeq12d 3708 . . . . . . . . . . . . . . 15 |- ( u = y -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q y } , { q | y <Q q } >. )
2625breq2d 3936 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 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 5417 . . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . . . . 16 |- Q. e. _V
3029rabex 4067 . . . . . . . . . . . . . . 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 4067 . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . 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 2158 . . . . . . . . . . . . 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 2838 . . . . . . . . . . . 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 119 . . . . . . . . . . 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 113 . . . . . . . . . 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 5414 . . . . . . . . . . . 12 |- ( r = b -> ( F ` r ) = ( F ` b ) )
39 opeq1 3700 . . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
4039eceq1d 6458 . . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
4140fveq2d 5418 . . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
4241breq2d 3936 . . . . . . . . . . . . . 14 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
4342abbidv 2255 . . . . . . . . . . . . 13 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
4441breq1d 3934 . . . . . . . . . . . . . 14 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
4544abbidv 2255 . . . . . . . . . . . . 13 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
4643, 45opeq12d 3708 . . . . . . . . . . . 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 5785 . . . . . . . . . . 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 3934 . . . . . . . . . 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 2653 . . . . . . . . 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 121 . . . . . . . 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 109 . . . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
5352adantl 275 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . . . 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 5549 . . . . . . . . . . . . . . . . 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 7349 . . . . . . . . . . . . . . . . . 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 7338 . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 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 111 . . . . . . . . . . . . . . . . . 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 7348 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 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 7379 . . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
6867adantl 275 . . . . . . . . . . . . . . . 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 5933 . . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 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 7362 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 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 3951 . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 3928 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( p <Q ( y +Q Q ) <-> p <Q x ) )
7877abbidv 2255 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { p | p <Q ( y +Q Q ) } = { p | p <Q x } )
79 breq1 3927 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( ( y +Q Q ) <Q q <-> x <Q q ) )
8079abbidv 2255 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { q | ( y +Q Q ) <Q q } = { q | x <Q q } )
8178, 80opeq12d 3708 . . . . . . . . . . . . . . 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 3936 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . 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 524 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 7338 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 7353 . . . . . . . . . . . . 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 408 . . . . . . . . . . . 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 166 . . . . . . . . . . 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 529 . . . . . . . . . . . 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 483 . . . . . . . . . . 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 304 . . . . . . . . . 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 114 . . . . . . . . 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 2532 . . . . . . . 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 2552 . . . . . 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 372 . . . . 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 2532 . . . 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 2593 . . 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 121 . 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 5548 . . . . . 6 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
10557adantl 275 . . . . . 6 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
106104, 105, 59syl2anc 408 . . . . 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 274 . . . . 5 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
108106, 107, 87syl2anc 408 . . . 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 274 . . . 4 |- ( ( ph /\ b e. N. ) -> T e. P. )
110 ltdfpr 7307 . . . 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 408 . . 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 2432 . 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 166 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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   Q.cnq 7081   +Q cplq 7083   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   +P. cpp 7094   <P cltp 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271
This theorem is referenced by:  caucvgprprlemexb  7508
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