Theorem caucvgprprlemloc 6893

Description: Lemma for caucvgprpr 6902. 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 6599	. . . . 5 |- ( s <Q t -> E. y e. Q. ( s +Q y ) = t )
2	1	adantl 271	. . . 4 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> E. y e. Q. ( s +Q y ) = t )
3		subhalfnqq 6604	. . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
4	3	ad2antrl 473	. . . . 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 6853	. . . . . . 7 |- ( x e. Q. -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
6	5	ad2antrl 473	. . . . . 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 500	. . . . . . . . . 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 270	. . . . . . . . 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 501	. . . . . . . . . 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 270	. . . . . . . . 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 502	. . . . . . . . . 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 270	. . . . . . . . 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 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 ) ) -> x e. Q. )
14		simplrr 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 ) ) -> ( x +Q x ) <Q y )
15		simprl 497	. . . . . . . . 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 498	. . . . . . . . 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 6874	. . . . . . . 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 501	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> s e. Q. )
19	18	ad3antrrr 475	. . . . . . . . . . . 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 6612	. . . . . . . . . . . . . 14 |- ( c e. N. -> [ <. c , 1o >. ] ~Q e. Q. )
21	20	ad2antrl 473	. . . . . . . . . . . . 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 6582	. . . . . . . . . . . . 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 6565	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
25	19, 23, 24	syl2anc 403	. . . . . . . . . . 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 6737	. . . . . . . . . . 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 6737	. . . . . . . . . . 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 6727	. . . . . . . . . 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 403	. . . . . . . . 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 502	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> t e. Q. )
33	32	ad3antrrr 475	. . . . . . . . . 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 6737	. . . . . . . . . 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 479	. . . . . . . . . . 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 5324	. . . . . . . . . 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 6555	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	39	brel 4410	. . . . . . . . . . . . 13 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) )
41	40	simpld 110	. . . . . . . . . . . 12 |- ( ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. )
42	41	ad2antll 474	. . . . . . . . . . 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 6727	. . . . . . . . . 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 403	. . . . . . . . 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 6786	. . . . . . . . . 10 |- <P Or P.
47		sowlin 4075	. . . . . . . . . 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 414	. . . . . . . . 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 1169	. . . . . . . 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 270	. . . . . . . . . 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 501	. . . . . . . . . . 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 108	. . . . . . . . . . . 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 6809	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
55	54	adantl 271	. . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . 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 270	. . . . . . . . . . . . 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 6768	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
62	61	adantl 271	. . . . . . . . . . . . 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 5690	. . . . . . . . . . . 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 165	. . . . . . . . . . 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 3570	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> <. a , 1o >. = <. c , 1o >. )
66	65	eceq1d 6165	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> [ <. a , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
67	66	fveq2d 5202	. . . . . . . . . . . . . . . . 17 |- ( a = c -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
68	67	oveq2d 5548	. . . . . . . . . . . . . . . 16 |- ( a = c -> ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
69	68	breq2d 3797	. . . . . . . . . . . . . . 15 |- ( a = c -> ( p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
70	69	abbidv 2196	. . . . . . . . . . . . . 14 |- ( a = c -> { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } )
71	68	breq1d 3795	. . . . . . . . . . . . . . 15 |- ( a = c -> ( ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q ) )
72	71	abbidv 2196	. . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . 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 5198	. . . . . . . . . . . . 13 |- ( a = c -> ( F ` a ) = ( F ` c ) )
75	73, 74	breq12d 3798	. . . . . . . . . . . 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 2701	. . . . . . . . . . 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 403	. . . . . . . . . 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 6875	. . . . . . . . . 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 408	. . . . . . . . 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 113	. . . . . . . 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 270	. . . . . . . . . 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 5198	. . . . . . . . . . . . . 14 |- ( b = c -> ( F ` b ) = ( F ` c ) )
84		opeq1 3570	. . . . . . . . . . . . . . . . . . 19 |- ( b = c -> <. b , 1o >. = <. c , 1o >. )
85	84	eceq1d 6165	. . . . . . . . . . . . . . . . . 18 |- ( b = c -> [ <. b , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
86	85	fveq2d 5202	. . . . . . . . . . . . . . . . 17 |- ( b = c -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
87	86	breq2d 3797	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
88	87	abbidv 2196	. . . . . . . . . . . . . . 15 |- ( b = c -> { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } )
89	86	breq1d 3795	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q ) )
90	89	abbidv 2196	. . . . . . . . . . . . . . 15 |- ( b = c -> { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } )
91	88, 90	opeq12d 3578	. . . . . . . . . . . . . 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 5550	. . . . . . . . . . . . 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 3795	. . . . . . . . . . . 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 2701	. . . . . . . . . . 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 277	. . . . . . . . . 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 6876	. . . . . . . . . 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 408	. . . . . . . . 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 113	. . . . . . . 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 732	. . . . . . 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 2481	. . . . 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 2481	. . . 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 2481	. . 3 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) )
104	103	ex 113	. 2 |- ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) -> ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
105	104	ralrimivva 2443	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 102   <-> wb 103   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   Or wor 4050   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   +P. cpp 6483   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660

This theorem is referenced by:  caucvgprprlemcl  6894