Theorem caucvgprprlemloc 7511

Description: Lemma for caucvgprpr 7520. 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 7217	. . . . 5 |- ( s <Q t -> E. y e. Q. ( s +Q y ) = t )
2	1	adantl 275	. . . 4 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> E. y e. Q. ( s +Q y ) = t )
3		subhalfnqq 7222	. . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
4	3	ad2antrl 481	. . . . 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 7471	. . . . . . 7 |- ( x e. Q. -> E. c e. N. ( *Q ` [ <. c , 1o >. ] ~Q ) <Q x )
6	5	ad2antrl 481	. . . . . 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 523	. . . . . . . . . 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 274	. . . . . . . . 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 524	. . . . . . . . . 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 274	. . . . . . . . 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 525	. . . . . . . . . 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 274	. . . . . . . . 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 524	. . . . . . . . 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 525	. . . . . . . . 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 520	. . . . . . . . 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 521	. . . . . . . . 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 7492	. . . . . . . 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 524	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> s e. Q. )
19	18	ad3antrrr 483	. . . . . . . . . . . 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 7230	. . . . . . . . . . . . . 14 |- ( c e. N. -> [ <. c , 1o >. ] ~Q e. Q. )
21	20	ad2antrl 481	. . . . . . . . . . . . 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 7200	. . . . . . . . . . . . 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 7183	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( *Q ` [ <. c , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) e. Q. )
25	19, 23, 24	syl2anc 408	. . . . . . . . . . 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 7355	. . . . . . . . . . 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 7355	. . . . . . . . . . 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 7345	. . . . . . . . . 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 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 } >. ) e. P. )
32		simplrr 525	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ t e. Q. ) ) /\ s <Q t ) -> t e. Q. )
33	32	ad3antrrr 483	. . . . . . . . . 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 7355	. . . . . . . . . 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 487	. . . . . . . . . . 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 5556	. . . . . . . . . 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 7173	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	39	brel 4591	. . . . . . . . . . . . 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 482	. . . . . . . . . . 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 7345	. . . . . . . . . 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 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 } >. ) e. P. )
46		ltsopr 7404	. . . . . . . . . 10 |- <P Or P.
47		sowlin 4242	. . . . . . . . . 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 420	. . . . . . . . 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 1216	. . . . . . . 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 274	. . . . . . . . . 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 524	. . . . . . . . . . 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 7427	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
55	54	adantl 275	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . 14 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
62	61	adantl 275	. . . . . . . . . . . . 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 5940	. . . . . . . . . . . 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 3705	. . . . . . . . . . . . . . . . . . 19 |- ( a = c -> <. a , 1o >. = <. c , 1o >. )
66	65	eceq1d 6465	. . . . . . . . . . . . . . . . . 18 |- ( a = c -> [ <. a , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
67	66	fveq2d 5425	. . . . . . . . . . . . . . . . 17 |- ( a = c -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
68	67	oveq2d 5790	. . . . . . . . . . . . . . . 16 |- ( a = c -> ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
69	68	breq2d 3941	. . . . . . . . . . . . . . 15 |- ( a = c -> ( p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
70	69	abbidv 2257	. . . . . . . . . . . . . 14 |- ( a = c -> { p | p <Q ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) } )
71	68	breq1d 3939	. . . . . . . . . . . . . . 15 |- ( a = c -> ( ( s +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <Q q ) )
72	71	abbidv 2257	. . . . . . . . . . . . . 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 3713	. . . . . . . . . . . . 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 5421	. . . . . . . . . . . . 13 |- ( a = c -> ( F ` a ) = ( F ` c ) )
75	73, 74	breq12d 3942	. . . . . . . . . . . 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 2789	. . . . . . . . . . 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 408	. . . . . . . . . 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 7493	. . . . . . . . . 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 413	. . . . . . . . 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 274	. . . . . . . . . 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 5421	. . . . . . . . . . . . . 14 |- ( b = c -> ( F ` b ) = ( F ` c ) )
84		opeq1 3705	. . . . . . . . . . . . . . . . . . 19 |- ( b = c -> <. b , 1o >. = <. c , 1o >. )
85	84	eceq1d 6465	. . . . . . . . . . . . . . . . . 18 |- ( b = c -> [ <. b , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
86	85	fveq2d 5425	. . . . . . . . . . . . . . . . 17 |- ( b = c -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
87	86	breq2d 3941	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
88	87	abbidv 2257	. . . . . . . . . . . . . . 15 |- ( b = c -> { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. c , 1o >. ] ~Q ) } )
89	86	breq1d 3939	. . . . . . . . . . . . . . . 16 |- ( b = c -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q ) )
90	89	abbidv 2257	. . . . . . . . . . . . . . 15 |- ( b = c -> { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. c , 1o >. ] ~Q ) <Q q } )
91	88, 90	opeq12d 3713	. . . . . . . . . . . . . 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 5792	. . . . . . . . . . . . 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 3939	. . . . . . . . . . . 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 2789	. . . . . . . . . . 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 281	. . . . . . . . . 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 7494	. . . . . . . . . 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 413	. . . . . . . . 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 775	. . . . . . 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 2554	. . . . 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 2554	. . . 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 2554	. . 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 2514	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 697   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   Or wor 4217   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7080   <N clti 7083   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093   P.cnp 7099   +P. cpp 7101   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  caucvgprprlemcl  7512