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Theorem caucvgprprlemaddq 7768
Description: Lemma for caucvgprpr 7772. 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 1539 . . 3 |- F/ r ph
3 nfcv 2336 . . . 4 |- F/_ r X
4 nfcv 2336 . . . 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 2537 . . . . . . . 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 2336 . . . . . . . 8 |- F/_ r Q.
86, 7nfrabw 2675 . . . . . . 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 2537 . . . . . . . 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 2675 . . . . . . 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 3820 . . . . . 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 2333 . . . . 5 |- F/_ r L
13 nfcv 2336 . . . . 5 |- F/_ r +P.
14 nfcv 2336 . . . . 5 |- F/_ r Q
1512, 13, 14nfov 5948 . . . 4 |- F/_ r ( L +P. Q )
163, 4, 15nfbr 4075 . . 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 535 . . . . . . . . . . 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 7753 . . . . . . . . . 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 5694 . . . . . . . . . . . . . 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 7608 . . . . . . . . . . . . . . 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 7597 . . . . . . . . . . . . . 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 7608 . . . . . . . . . . . . . 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 7639 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 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 4039 . . . . . . . . . . 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 7679 . . . . . . . . . . . . 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 7597 . . . . . . . . . . . . 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 5694 . . . . . . . . . . . 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 7638 . . . . . . . . . . . . 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 6088 . . . . . . . . . . 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 7638 . . . . . . . . . . . . . 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 5934 . . . . . . . . . . . 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 4039 . . . . . . . . . . 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 674 . . . . . . . . 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 2587 . . . . . . . 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 4032 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
5251cbvabv 2318 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) }
53 breq2 4033 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u ) )
5453cbvabv 2318 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u }
5552, 54opeq12i 3809 . . . . . . . . . . . 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 5929 . . . . . . . . . . 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 4032 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
5857cbvabv 2318 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
59 breq2 4033 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u ) )
6059cbvabv 2318 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u }
6158, 60opeq12i 3809 . . . . . . . . . . . 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 5929 . . . . . . . . . . 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 5930 . . . . . . . . . 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 4036 . . . . . . . . 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 2501 . . . . . . . 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 678 . . . . . . 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 529 . . . . . . . 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 7767 . . . . . . 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 664 . . . . . 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 531 . . . . . . 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 7608 . . . . . . . . . 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 7597 . . . . . . . . 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 5694 . . . . . . . . 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 7597 . . . . . . . . 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 7764 . . . . . . . . . . 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 7597 . . . . . . . . . 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 7597 . . . . . . . . 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 7656 . . . . . . . . 9 |- <P Or P.
92 sowlin 4351 . . . . . . . . 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 1249 . . . . . . 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 1360 . . . . 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 6088 . . . . 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 2608 . 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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   Or wor 4326   -->wf 5250   ` cfv 5254  (class class class)co 5918   1oc1o 6462   [cec 6585   N.cnpi 7332   <N clti 7335   ~Q ceq 7339   Q.cnq 7340   +Q cplq 7342   *Qcrq 7344   <Q cltq 7345   P.cnp 7351   +P. cpp 7353   <P cltp 7355
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530
This theorem is referenced by:  caucvgprprlem1  7769
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