Theorem caucvgprprlemloc 7922

Description: Lemma for caucvgprpr 7931. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)

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 } >. } >.

Assertion

Ref	Expression
caucvgprprlemloc	|- ( ph -> A. s e. Q. A. t e. Q. ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )

Distinct variable groups:   A, m   m, F   F, l, r   u, F, r   q, p, s, t   ph, s, t   p, l, q, s, t, r   u, p, q, s, t

Allowed substitution hints:   ph( u, k, m, n, r, q, p, l)   A( u, t, k, n, s, r, q, p, l)   F( t, k, n, s, q, p)   L( u, t, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemloc

Dummy variables a b f g h c x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7628	. . . . 5 |- ( s <Q t -> E. y e. Q. ( s +Q y ) = t )
2	1	adantl 277	. . . 4 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> E. y e. Q. ( s +Q y ) = t )
3		subhalfnqq 7633	. . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
4	3	ad2antrl 490	. . . . 5 |- ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) -> E. x e. Q. ( x +Q x ) <Q y )
5		archrecnq 7882	. . . . . . 7 |- ( x e. Q. -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
6	5	ad2antrl 490	. . . . . 6 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
7		simpllr 536	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> s <Q t )
8	7	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> s <Q t )
9		simplrl 537	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> y e. Q. )
10	9	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> y e. Q. )
11		simplrr 538	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> ( s +Q y ) = t )
12	11	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q y ) = t )
13		simplrl 537	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> x e. Q. )
14		simplrr 538	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( x +Q x ) <Q y )
15		simprl 531	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> c e. N. )
16		simprr 533	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
17	8, 10, 12, 13, 14, 15, 16	caucvgprprlemloccalc 7903	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
18		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> s e. Q. )
19	18	ad3antrrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> s e. Q. )
20		nnnq 7641	. . . . . . . . . . . . . 14 |- ( c e. N. -> [ <. c , 1o >. ] ~Q e. Q. )
21	20	ad2antrl 490	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> [ <. c , 1o >. ] ~Q e. Q. )
22		recclnq 7611	. . . . . . . . . . . . 13 |- ( [ <. c , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
23	21, 22	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
24		addclnq 7594	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
25	19, 23, 24	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
26		nqprlu 7766	. . . . . . . . . . 11 |- ( ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
27	25, 26	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
28		nqprlu 7766	. . . . . . . . . . 11 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
29	23, 28	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
30		addclpr 7756	. . . . . . . . . 10 |- ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. /\ <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
31	27, 29, 30	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
32		simplrr 538	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> t e. Q. )
33	32	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> t e. Q. )
34		nqprlu 7766	. . . . . . . . . 10 |- ( t e. Q. -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
35	33, 34	syl 14	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
36		caucvgprpr.f	. . . . . . . . . . . 12 |- ( ph -> F : N. --> P. )
37	36	ad5antr 496	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> F : N. --> P. )
38	37, 15	ffvelcdmd 5783	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( F ` c ) e. P. )
39		ltrelnq 7584	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	39	brel 4778	. . . . . . . . . . . . 13 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) )
41	40	simpld 112	. . . . . . . . . . . 12 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
42	41	ad2antll 491	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
43	42, 28	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
44		addclpr 7756	. . . . . . . . . 10 |- ( ( ( F ` c ) e. P. /\ <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
45	38, 43, 44	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. )
46		ltsopr 7815	. . . . . . . . . 10 |- <P Or P.
47		sowlin 4417	. . . . . . . . . 10 |- ( ( <P Or P. /\ ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q t } , { q | t <Q q } >. e. P. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
48	46, 47	mpan 424	. . . . . . . . 9 |- ( ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q t } , { q | t <Q q } >. e. P. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
49	31, 35, 45, 48	syl3anc 1273	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) )
50	17, 49	mpd 13	. . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
51	19	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> s e. Q. )
52		simplrl 537	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> c e. N. )
53		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) )
54		ltaprg 7838	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
55	54	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
56	42	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
57	51, 56, 24	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
58	57, 26	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. e. P. )
59	38	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( F ` c ) e. P. )
60	56, 28	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. e. P. )
61		addcomprg 7797	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
62	61	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
63	55, 58, 59, 60, 62	caovord2d 6191	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) <-> ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) )
64	53, 63	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) )
65		opeq1 3862	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> <. a , 1o >. = <. c , 1o >. )
66	65	eceq1d 6737	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> [ <. a , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
67	66	fveq2d 5643	. . . . . . . . . . . . . . . . 17 |- ( a = c -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
68	67	oveq2d 6033	. . . . . . . . . . . . . . . 16 |- ( a = c -> ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
69	68	breq2d 4100	. . . . . . . . . . . . . . 15 |- ( a = c -> ( p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
70	69	abbidv 2349	. . . . . . . . . . . . . 14 |- ( a = c -> { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } )
71	68	breq1d 4098	. . . . . . . . . . . . . . 15 |- ( a = c -> ( ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q ) )
72	71	abbidv 2349	. . . . . . . . . . . . . 14 |- ( a = c -> { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } = { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } )
73	70, 72	opeq12d 3870	. . . . . . . . . . . . 13 |- ( a = c -> <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. )
74		fveq2 5639	. . . . . . . . . . . . 13 |- ( a = c -> ( F ` a ) = ( F ` c ) )
75	73, 74	breq12d 4101	. . . . . . . . . . . 12 |- ( a = c -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) ) )
76	75	rspcev 2910	. . . . . . . . . . 11 |- ( ( c e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` c ) ) -> E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) )
77	52, 64, 76	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) )
78		caucvgprpr.lim	. . . . . . . . . . 11 |- 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 } >. } >.
79	78	caucvgprprlemell 7904	. . . . . . . . . 10 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. a e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) ) )
80	51, 77, 79	sylanbrc 417	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) ) -> s e. ( 1st ` L ) )
81	80	ex 115	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) -> s e. ( 1st ` L ) ) )
82	33	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> t e. Q. )
83		fveq2 5639	. . . . . . . . . . . . . 14 |- ( b = c -> ( F ` b ) = ( F ` c ) )
84		opeq1 3862	. . . . . . . . . . . . . . . . . . 19 |- ( b = c -> <. b , 1o >. = <. c , 1o >. )
85	84	eceq1d 6737	. . . . . . . . . . . . . . . . . 18 |- ( b = c -> [ <. b , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
86	85	fveq2d 5643	. . . . . . . . . . . . . . . . 17 |- ( b = c -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
87	86	breq2d 4100	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
88	87	abbidv 2349	. . . . . . . . . . . . . . 15 |- ( b = c -> { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } )
89	86	breq1d 4098	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q ) )
90	89	abbidv 2349	. . . . . . . . . . . . . . 15 |- ( b = c -> { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } )
91	88, 90	opeq12d 3870	. . . . . . . . . . . . . 14 |- ( b = c -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. )
92	83, 91	oveq12d 6035	. . . . . . . . . . . . 13 |- ( b = c -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) )
93	92	breq1d 4098	. . . . . . . . . . . 12 |- ( b = c -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. <-> ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
94	93	rspcev 2910	. . . . . . . . . . 11 |- ( ( c e. N. /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
95	15, 94	sylan 283	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
96	78	caucvgprprlemelu 7905	. . . . . . . . . 10 |- ( t e. ( 2nd ` L ) <-> ( t e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
97	82, 95, 96	sylanbrc 417	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> t e. ( 2nd ` L ) )
98	97	ex 115	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. -> t e. ( 2nd ` L ) ) )
99	81, 98	orim12d 793	. . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) \/ ( ( F ` c ) +P. <. { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
100	50, 99	mpd 13	. . . . . 6 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( c e. N. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
101	6, 100	rexlimddv 2655	. . . . 5 |- ( ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
102	4, 101	rexlimddv 2655	. . . 4 |- ( ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) /\ ( y e. Q. /\ ( s +Q y ) = t ) ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
103	2, 102	rexlimddv 2655	. . 3 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
104	103	ex 115	. 2 |- ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) -> ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
105	104	ralrimivva 2614	1 |- ( ph -> A. s e. Q. A. t e. Q. ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   {crab 2514   <.cop 3672   class class class wbr 4088   Or wor 4392   -->wf 5322   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   1oc1o 6574   [cec 6699   N.cnpi 7491   <N clti 7494   ~Q ceq 7498   Q.cnq 7499   +Q cplq 7501   *Qcrq 7503   <Q cltq 7504   P.cnp 7510   +P. cpp 7512   <P cltp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iplp 7687  df-iltp 7689

This theorem is referenced by:  caucvgprprlemcl  7923