ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemaddq Unicode version
Theorem caucvgprprlemaddq 8023
Description: Lemma for caucvgprpr 8027. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-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 } >. } >.
caucvgprprlemaddq.x |- ( ph -> X e. P. )
caucvgprprlemaddq.q |- ( ph -> Q e. P. )
caucvgprprlemaddq.ex |- ( ph -> E. r e. N. ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
Assertion
Ref Expression
caucvgprprlemaddq |- ( ph -> X <P ( L +P. Q ) )
Distinct variable groups:   A, m   m, F   A, r, m   F, l, r, u, k, n   k, L   Q, r   X, r   p, l, q, r, u   ph, r   k, p, q
Allowed substitution hints:   ph( u, k, m, n, q, p, l)   A( u, k, n, q, p, l)   Q( u, k, m, n, q, p, l)   F( q, p)   L( u, m, n, r, q, p, l)   X( u, k, m, n, q, p, l)
Proof of Theorem caucvgprprlemaddq
Dummy variables b f g h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemaddq.ex . 2 |- ( ph -> E. r e. N. ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
2 nfv 1577 . . 3 |- F/ r ph
3 nfcv 2384 . . . 4 |- F/_ r X
4 nfcv 2384 . . . 4 |- F/_ r <P
5 caucvgprpr.lim . . . . . 6 |- 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 } >. } >.
6 nfre1 2585 . . . . . . . 8 |- F/ r 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 )
7 nfcv 2384 . . . . . . . 8 |- F/_ r Q.
86, 7nfrabw 2725 . . . . . . 7 |- F/_ r { 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 ) }
9 nfre1 2585 . . . . . . . 8 |- F/ r 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 } >.
109, 7nfrabw 2725 . . . . . . 7 |- 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 } >. }
118, 10nfop 3899 . . . . . 6 |- F/_ r <. { 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 } >. } >.
125, 11nfcxfr 2381 . . . . 5 |- F/_ r L
13 nfcv 2384 . . . . 5 |- F/_ r +P.
14 nfcv 2384 . . . . 5 |- F/_ r Q
1512, 13, 14nfov 6080 . . . 4 |- F/_ r ( L +P. Q )
163, 4, 15nfbr 4156 . . 3 |- F/ r X <P ( L +P. Q )
17 caucvgprpr.f . . . . . . . . . . . 12 |- ( ph -> F : N. --> P. )
1817ad2antrr 488 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> F : N. --> P. )
19 caucvgprpr.cau . . . . . . . . . . . 12 |- ( 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 } >. ) ) ) )
2019ad2antrr 488 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> 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 } >. ) ) ) )
21 simpr 110 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> b e. N. )
22 simplrl 537 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> r e. N. )
2318, 20, 21, 22caucvgprprlemnbj 8008 . . . . . . . . . 10 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> -. ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` r ) )
2418, 21ffvelcdmd 5813 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( F ` b ) e. P. )
25 recnnpr 7863 . . . . . . . . . . . . . . 15 |- ( b e. N. -> <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. e. P. )
2625adantl 277 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. e. P. )
27 addclpr 7852 . . . . . . . . . . . . . 14 |- ( ( ( F ` b ) e. P. /\ <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) e. P. )
2824, 26, 27syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) e. P. )
29 recnnpr 7863 . . . . . . . . . . . . . 14 |- ( r e. N. -> <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. e. P. )
3022, 29syl 14 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. e. P. )
31 caucvgprprlemaddq.q . . . . . . . . . . . . . 14 |- ( ph -> Q e. P. )
3231ad2antrr 488 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> Q e. P. )
33 addassprg 7894 . . . . . . . . . . . . 13 |- ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) e. P. /\ <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. e. P. /\ Q e. P. ) -> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) +P. Q ) = ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) ) )
3428, 30, 32, 33syl3anc 1274 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) +P. Q ) = ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) ) )
3534breq1d 4119 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) +P. Q ) <P ( ( F ` r ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) ) <P ( ( F ` r ) +P. Q ) ) )
36 ltaprg 7934 . . . . . . . . . . . . 13 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
3736adantl 277 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
38 addclpr 7852 . . . . . . . . . . . . 13 |- ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) e. P. /\ <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) e. P. )
3928, 30, 38syl2anc 411 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) e. P. )
4018, 22ffvelcdmd 5813 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( F ` r ) e. P. )
41 addcomprg 7893 . . . . . . . . . . . . 13 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
4241adantl 277 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
4337, 39, 40, 32, 42caovord2d 6224 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` r ) <-> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) +P. Q ) <P ( ( F ` r ) +P. Q ) ) )
44 addcomprg 7893 . . . . . . . . . . . . . 14 |- ( ( Q e. P. /\ <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) = ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) )
4532, 30, 44syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) = ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) )
4645oveq2d 6066 . . . . . . . . . . . 12 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) = ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) ) )
4746breq1d 4119 . . . . . . . . . . 11 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. +P. Q ) ) <P ( ( F ` r ) +P. Q ) ) )
4835, 43, 473bitr4rd 221 . . . . . . . . . 10 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` r ) ) )
4923, 48mtbird 680 . . . . . . . . 9 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ b e. N. ) -> -. ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) )
5049nrexdv 2635 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> -. E. b e. N. ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) )
51 breq1 4112 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
5251cbvabv 2359 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) }
53 breq2 4113 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u ) )
5453cbvabv 2359 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u }
5552, 54opeq12i 3888 . . . . . . . . . . . 12 |- <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. = <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >.
5655oveq2i 6061 . . . . . . . . . . 11 |- ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. )
57 breq1 4112 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
5857cbvabv 2359 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
59 breq2 4113 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u ) )
6059cbvabv 2359 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u }
6158, 60opeq12i 3888 . . . . . . . . . . . 12 |- <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >.
6261oveq2i 6061 . . . . . . . . . . 11 |- ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. )
6356, 62oveq12i 6062 . . . . . . . . . 10 |- ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) = ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) )
6463breq1i 4116 . . . . . . . . 9 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` r ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) )
6564rexbii 2549 . . . . . . . 8 |- ( E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` r ) +P. Q ) <-> E. b e. N. ( ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. ) +P. ( Q +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) ) <P ( ( F ` r ) +P. Q ) )
6650, 65sylnibr 684 . . . . . . 7 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> -. E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` r ) +P. Q ) )
6717adantr 276 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> F : N. --> P. )
6819adantr 276 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> 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 } >. ) ) ) )
69 caucvgprpr.bnd . . . . . . . . 9 |- ( ph -> A. m e. N. A <P ( F ` m ) )
7069adantr 276 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> A. m e. N. A <P ( F ` m ) )
7131adantr 276 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> Q e. P. )
72 simprl 531 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> r e. N. )
7367, 68, 70, 5, 71, 72caucvgprprlemexb 8022 . . . . . . 7 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` r ) +P. Q ) ) )
7466, 73mtod 669 . . . . . 6 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> -. ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
75 simprr 533 . . . . . . 7 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
76 caucvgprprlemaddq.x . . . . . . . . . 10 |- ( ph -> X e. P. )
7776adantr 276 . . . . . . . . 9 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> X e. P. )
78 recnnpr 7863 . . . . . . . . . 10 |- ( r e. N. -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. e. P. )
7972, 78syl 14 . . . . . . . . 9 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. e. P. )
80 addclpr 7852 . . . . . . . . 9 |- ( ( X e. P. /\ <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. )
8177, 79, 80syl2anc 411 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. )
8267, 72ffvelcdmd 5813 . . . . . . . . 9 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( F ` r ) e. P. )
83 addclpr 7852 . . . . . . . . 9 |- ( ( ( F ` r ) e. P. /\ Q e. P. ) -> ( ( F ` r ) +P. Q ) e. P. )
8482, 71, 83syl2anc 411 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( ( F ` r ) +P. Q ) e. P. )
8517, 19, 69, 5caucvgprprlemcl 8019 . . . . . . . . . . 11 |- ( ph -> L e. P. )
8685adantr 276 . . . . . . . . . 10 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> L e. P. )
87 addclpr 7852 . . . . . . . . . 10 |- ( ( L e. P. /\ Q e. P. ) -> ( L +P. Q ) e. P. )
8886, 71, 87syl2anc 411 . . . . . . . . 9 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( L +P. Q ) e. P. )
89 addclpr 7852 . . . . . . . . 9 |- ( ( ( L +P. Q ) e. P. /\ <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. )
9088, 79, 89syl2anc 411 . . . . . . . 8 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. )
91 ltsopr 7911 . . . . . . . . 9 |- <P Or P.
92 sowlin 4441 . . . . . . . . 9 |- ( ( <P Or P. /\ ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ ( ( F ` r ) +P. Q ) e. P. /\ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. ) ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) )
9391, 92mpan 424 . . . . . . . 8 |- ( ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ ( ( F ` r ) +P. Q ) e. P. /\ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) )
9481, 84, 90, 93syl3anc 1274 . . . . . . 7 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) )
9575, 94mpd 13 . . . . . 6 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) )
9674, 95ecased 1386 . . . . 5 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) )
9736adantl 277 . . . . . 6 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
9841adantl 277 . . . . . 6 |- ( ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
9997, 77, 88, 79, 98caovord2d 6224 . . . . 5 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> ( X <P ( L +P. Q ) <-> ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) ) )
10096, 99mpbird 167 . . . 4 |- ( ( ph /\ ( r e. N. /\ ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) ) ) -> X <P ( L +P. Q ) )
101100exp32 365 . . 3 |- ( ph -> ( r e. N. -> ( ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> X <P ( L +P. Q ) ) ) )
1022, 16, 101rexlimd 2657 . 2 |- ( ph -> ( E. r e. N. ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) -> X <P ( L +P. Q ) ) )
1031, 102mpd 13 1 |- ( ph -> X <P ( L +P. Q ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3692   class class class wbr 4109   Or wor 4416   -->wf 5348   ` cfv 5352  (class class class)co 6050   1oc1o 6640   [cec 6765   N.cnpi 7587   <N clti 7590   ~Q ceq 7594   Q.cnq 7595   +Q cplq 7597   *Qcrq 7599   <Q cltq 7600   P.cnp 7606   +P. cpp 7608   <P cltp 7610
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785
This theorem is referenced by:  caucvgprprlem1  8024
  Copyright terms: Public domain W3C validator