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Theorem caucvgprprlemexbt 7707
Description: Lemma for caucvgprpr 7713. 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 7706 . . . . . . 7 |- ( ph -> L e. P. )
7 caucvgprprlemexbt.q . . . . . . . 8 |- ( ph -> Q e. Q. )
8 nqprlu 7548 . . . . . . . 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 7538 . . . . . . 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 7507 . . . . . 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 7621 . . . . . . 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 4009 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( p <Q u <-> p <Q y ) )
2221abbidv 2295 . . . . . . . . . . . . . . . 16 |- ( u = y -> { p | p <Q u } = { p | p <Q y } )
23 breq1 4008 . . . . . . . . . . . . . . . . 17 |- ( u = y -> ( u <Q q <-> y <Q q ) )
2423abbidv 2295 . . . . . . . . . . . . . . . 16 |- ( u = y -> { q | u <Q q } = { q | y <Q q } )
2522, 24opeq12d 3788 . . . . . . . . . . . . . . 15 |- ( u = y -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q y } , { q | y <Q q } >. )
2625breq2d 4017 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . 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 5520 . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . 16 |- Q. e. _V
3029rabex 4149 . . . . . . . . . . . . . . 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 4149 . . . . . . . . . . . . . . 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 6150 . . . . . . . . . . . . . 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 2198 . . . . . . . . . . . . 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 2898 . . . . . . . . . . . 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 5517 . . . . . . . . . . . 12 |- ( r = b -> ( F ` r ) = ( F ` b ) )
39 opeq1 3780 . . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
4039eceq1d 6573 . . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
4140fveq2d 5521 . . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
4241breq2d 4017 . . . . . . . . . . . . . 14 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
4342abbidv 2295 . . . . . . . . . . . . 13 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
4441breq1d 4015 . . . . . . . . . . . . . 14 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
4544abbidv 2295 . . . . . . . . . . . . 13 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
4643, 45opeq12d 3788 . . . . . . . . . . . 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 5895 . . . . . . . . . . 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 4015 . . . . . . . . . 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 2706 . . . . . . . . 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 7620 . . . . . . . . . . . . . . . . 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 5654 . . . . . . . . . . . . . . . . 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 7549 . . . . . . . . . . . . . . . . . 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 7538 . . . . . . . . . . . . . . . . 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 7548 . . . . . . . . . . . . . . . . 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 7579 . . . . . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . 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 4009 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( p <Q ( y +Q Q ) <-> p <Q x ) )
7877abbidv 2295 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { p | p <Q ( y +Q Q ) } = { p | p <Q x } )
79 breq1 4008 . . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( ( y +Q Q ) <Q q <-> x <Q q ) )
8079abbidv 2295 . . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { q | ( y +Q Q ) <Q q } = { q | x <Q q } )
8178, 80opeq12d 3788 . . . . . . . . . . . . . . 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 4017 . . . . . . . . . . . . . 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 7538 . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . 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 2579 . . . . . . . 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 2599 . . . . . 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 2579 . . . 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 2641 . . 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 5653 . . . . . 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 7507 . . . 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 2474 . 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 978   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   Q.cnq 7281   +Q cplq 7283   *Qcrq 7285   <Q cltq 7286   P.cnp 7292   +P. cpp 7294   <P cltp 7296
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471
This theorem is referenced by:  caucvgprprlemexb  7708
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