Theorem caucvgprprlemloc 7412

Description: Lemma for caucvgprpr 7421. 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 7118	. . . . 5 |- ( s <Q t -> E. y e. Q. ( s +Q y ) = t )
2	1	adantl 273	. . . 4 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> E. y e. Q. ( s +Q y ) = t )
3		subhalfnqq 7123	. . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
4	3	ad2antrl 477	. . . . 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 7372	. . . . . . 7 |- ( x e. Q. -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
6	5	ad2antrl 477	. . . . . 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 504	. . . . . . . . . 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 272	. . . . . . . . 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 505	. . . . . . . . . 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 272	. . . . . . . . 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 506	. . . . . . . . . 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 272	. . . . . . . . 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 505	. . . . . . . . 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 506	. . . . . . . . 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 501	. . . . . . . . 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 502	. . . . . . . . 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 7393	. . . . . . . 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 505	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> s e. Q. )
19	18	ad3antrrr 479	. . . . . . . . . . . 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 7131	. . . . . . . . . . . . . 14 |- ( c e. N. -> [ <. c , 1o >. ] ~Q e. Q. )
21	20	ad2antrl 477	. . . . . . . . . . . . 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 7101	. . . . . . . . . . . . 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 7084	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
25	19, 23, 24	syl2anc 406	. . . . . . . . . . 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 7256	. . . . . . . . . . 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 7256	. . . . . . . . . . 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 7246	. . . . . . . . . 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 406	. . . . . . . . 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 506	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> t e. Q. )
33	32	ad3antrrr 479	. . . . . . . . . 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 7256	. . . . . . . . . 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 483	. . . . . . . . . . 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	ffvelrnd 5488	. . . . . . . . . 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 7074	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	39	brel 4529	. . . . . . . . . . . . 13 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) )
41	40	simpld 111	. . . . . . . . . . . 12 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
42	41	ad2antll 478	. . . . . . . . . . 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 7246	. . . . . . . . . 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 406	. . . . . . . . 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 7305	. . . . . . . . . 10 |- <P Or P.
47		sowlin 4180	. . . . . . . . . 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 418	. . . . . . . . 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 1184	. . . . . . . 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 272	. . . . . . . . . 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 505	. . . . . . . . . . 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 109	. . . . . . . . . . . 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 7328	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
55	54	adantl 273	. . . . . . . . . . . . 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 272	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 272	. . . . . . . . . . . . 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 7287	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
62	61	adantl 273	. . . . . . . . . . . . 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 5872	. . . . . . . . . . . 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 166	. . . . . . . . . . 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 3652	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> <. a , 1o >. = <. c , 1o >. )
66	65	eceq1d 6395	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> [ <. a , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
67	66	fveq2d 5357	. . . . . . . . . . . . . . . . 17 |- ( a = c -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
68	67	oveq2d 5722	. . . . . . . . . . . . . . . 16 |- ( a = c -> ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
69	68	breq2d 3887	. . . . . . . . . . . . . . 15 |- ( a = c -> ( p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
70	69	abbidv 2217	. . . . . . . . . . . . . 14 |- ( a = c -> { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } )
71	68	breq1d 3885	. . . . . . . . . . . . . . 15 |- ( a = c -> ( ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q ) )
72	71	abbidv 2217	. . . . . . . . . . . . . 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 3660	. . . . . . . . . . . . 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 5353	. . . . . . . . . . . . 13 |- ( a = c -> ( F ` a ) = ( F ` c ) )
75	73, 74	breq12d 3888	. . . . . . . . . . . 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 2744	. . . . . . . . . . 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 406	. . . . . . . . . 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 7394	. . . . . . . . . 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 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 } >. ) <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 114	. . . . . . . 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 272	. . . . . . . . . 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 5353	. . . . . . . . . . . . . 14 |- ( b = c -> ( F ` b ) = ( F ` c ) )
84		opeq1 3652	. . . . . . . . . . . . . . . . . . 19 |- ( b = c -> <. b , 1o >. = <. c , 1o >. )
85	84	eceq1d 6395	. . . . . . . . . . . . . . . . . 18 |- ( b = c -> [ <. b , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
86	85	fveq2d 5357	. . . . . . . . . . . . . . . . 17 |- ( b = c -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
87	86	breq2d 3887	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
88	87	abbidv 2217	. . . . . . . . . . . . . . 15 |- ( b = c -> { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } )
89	86	breq1d 3885	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q ) )
90	89	abbidv 2217	. . . . . . . . . . . . . . 15 |- ( b = c -> { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } )
91	88, 90	opeq12d 3660	. . . . . . . . . . . . . 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 5724	. . . . . . . . . . . . 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 3885	. . . . . . . . . . . 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 2744	. . . . . . . . . . 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 279	. . . . . . . . . 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 7395	. . . . . . . . . 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 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 } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) -> t e. ( 2nd ` L ) )
98	97	ex 114	. . . . . . . 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 741	. . . . . . 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 2513	. . . . 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 2513	. . . 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 2513	. . 3 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
104	103	ex 114	. 2 |- ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) -> ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
105	104	ralrimivva 2473	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 103   <-> wb 104   \/ wo 670   /\ w3a 930   = wceq 1299   e. wcel 1448   {cab 2086   A.wral 2375   E.wrex 2376   {crab 2379   <.cop 3477   class class class wbr 3875   Or wor 4155   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   2ndc2nd 5968   1oc1o 6236   [cec 6357   N.cnpi 6981   <N clti 6984   ~Q ceq 6988   Q.cnq 6989   +Q cplq 6991   *Qcrq 6993   <Q cltq 6994   P.cnp 7000   +P. cpp 7002   <P cltp 7004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-iltp 7179

This theorem is referenced by:  caucvgprprlemcl  7413