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Theorem caucvgprprlemaddq 6806
Description: Lemma for caucvgprpr 6810. 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 1421 . . 3 |- F/ r ph
3 nfcv 2178 . . . 4 |- F/_ r X
4 nfcv 2178 . . . 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 2365 . . . . . . . 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 2178 . . . . . . . 8 |- F/_ r Q.
86, 7nfrabxy 2490 . . . . . . 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 2365 . . . . . . . 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, 7nfrabxy 2490 . . . . . . 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 3565 . . . . . 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 2175 . . . . 5 |- F/_ r L
13 nfcv 2178 . . . . 5 |- F/_ r +P.
14 nfcv 2178 . . . . 5 |- F/_ r Q
1512, 13, 14nfov 5535 . . . 4 |- F/_ r ( L +P. Q )
163, 4, 15nfbr 3808 . . 3 |- F/ r X <P ( L +P. Q )
17 caucvgprpr.f . . . . . . . . . . . 12 |- ( ph -> F : N. --> P. )
1817ad2antrr 457 . . . . . . . . . . 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 457 . . . . . . . . . . 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 103 . . . . . . . . . . 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 487 . . . . . . . . . . 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 6791 . . . . . . . . . 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, 21ffvelrnd 5303 . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . 15 |- ( b e. N. -> <. { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u } >. e. P. )
2625adantl 262 . . . . . . . . . . . . . 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 6635 . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 6677 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 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 3774 . . . . . . . . . . 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 6717 . . . . . . . . . . . . 13 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
3736adantl 262 . . . . . . . . . . . 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 6635 . . . . . . . . . . . . 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 391 . . . . . . . . . . . 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, 22ffvelrnd 5303 . . . . . . . . . . . 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 6676 . . . . . . . . . . . . 13 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
4241adantl 262 . . . . . . . . . . . 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 5670 . . . . . . . . . . 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 6676 . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . 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 5528 . . . . . . . . . . . 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 3774 . . . . . . . . . . 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 210 . . . . . . . . . 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 598 . . . . . . . . 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 2412 . . . . . . . 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 3767 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
5251cbvabv 2161 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. b , 1o >. ] ~Q ) }
53 breq2 3768 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u ) )
5453cbvabv 2161 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q u }
5552, 54opeq12i 3554 . . . . . . . . . . . 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 5523 . . . . . . . . . . 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 3767 . . . . . . . . . . . . . 14 |- ( p = l -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
5857cbvabv 2161 . . . . . . . . . . . . 13 |- { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
59 breq2 3768 . . . . . . . . . . . . . 14 |- ( q = u -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u ) )
6059cbvabv 2161 . . . . . . . . . . . . 13 |- { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u }
6158, 60opeq12i 3554 . . . . . . . . . . . 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 5523 . . . . . . . . . . 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 5524 . . . . . . . . . 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 3771 . . . . . . . . 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 2331 . . . . . . . 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 602 . . . . . . 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 261 . . . . . . . 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 261 . . . . . . . 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 261 . . . . . . . 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 261 . . . . . . . 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 483 . . . . . . . 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 6805 . . . . . . 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 589 . . . . . 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 484 . . . . . . 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 261 . . . . . . . . 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 6646 . . . . . . . . . 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 6635 . . . . . . . . 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 391 . . . . . . . 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, 72ffvelrnd 5303 . . . . . . . . 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 6635 . . . . . . . . 9 |- ( ( ( F ` r ) e. P. /\ Q e. P. ) -> ( ( F ` r ) +P. Q ) e. P. )
8482, 71, 83syl2anc 391 . . . . . . . 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 6802 . . . . . . . . . . 11 |- ( ph -> L e. P. )
8685adantr 261 . . . . . . . . . 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 6635 . . . . . . . . . 10 |- ( ( L e. P. /\ Q e. P. ) -> ( L +P. Q ) e. P. )
8886, 71, 87syl2anc 391 . . . . . . . . 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 6635 . . . . . . . . 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 391 . . . . . . . 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 6694 . . . . . . . . 9 |- <P Or P.
92 sowlin 4057 . . . . . . . . 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 400 . . . . . . . 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 1135 . . . . . . 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 1239 . . . . 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 262 . . . . . 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 262 . . . . . 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 5670 . . . . 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 156 . . . 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 347 . . 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 2430 . 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 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   Or wor 4032   -->wf 4898   ` cfv 4902  (class class class)co 5512   1oc1o 5994   [cec 6104   N.cnpi 6370   <N clti 6373   ~Q ceq 6377   Q.cnq 6378   +Q cplq 6380   *Qcrq 6382   <Q cltq 6383   P.cnp 6389   +P. cpp 6391   <P cltp 6393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568
This theorem is referenced by:  caucvgprprlem1  6807
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