Theorem caucvgprprlemopl 8028

Description: Lemma for caucvgprpr 8043. The lower cut of the putative limit is open. (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
caucvgprprlemopl	|- ( ( ph /\ s e. ( 1st ` L ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )

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

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

Proof of Theorem caucvgprprlemopl

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.lim	. . . . 5 |- 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 } >. } >.
2	1	caucvgprprlemell 8016	. . . 4 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
3	2	simprbi 275	. . 3 |- ( s e. ( 1st ` L ) -> E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
4	3	adantl 277	. 2 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
5		caucvgprpr.f	. . . . . . 7 |- ( ph -> F : N. --> P. )
6	5	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> F : N. --> P. )
7		simprl 531	. . . . . 6 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> b e. N. )
8	6, 7	ffvelcdmd 5818	. . . . 5 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> ( F ` b ) e. P. )
9		prop 7806	. . . . 5 |- ( ( F ` b ) e. P. -> <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. )
10	8, 9	syl 14	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. )
11		simprr 533	. . . . 5 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
12	1	caucvgprprlemell 8016	. . . . . . . . 9 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
13	12	simplbi 274	. . . . . . . 8 |- ( s e. ( 1st ` L ) -> s e. Q. )
14	13	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> s e. Q. )
15		nnnq 7753	. . . . . . . . 9 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
16		recclnq 7723	. . . . . . . . 9 |- ( [ <. b , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
17	15, 16	syl 14	. . . . . . . 8 |- ( b e. N. -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
18	17	ad2antrl 490	. . . . . . 7 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
19		addclnq 7706	. . . . . . 7 |- ( ( s e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
20	14, 18, 19	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
21		nqprl 7882	. . . . . 6 |- ( ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. /\ ( F ` b ) e. P. ) -> ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
22	20, 8, 21	syl2anc 411	. . . . 5 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
23	11, 22	mpbird 167	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) )
24		prnmaxl 7819	. . . 4 |- ( ( <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) ) -> E. a e. ( 1st ` ( F ` b ) ) ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a )
25	10, 23, 24	syl2anc 411	. . 3 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> E. a e. ( 1st ` ( F ` b ) ) ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a )
26	18	adantr 276	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
27	14	adantr 276	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> s e. Q. )
28		ltaddnq 7738	. . . . . . . 8 |- ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. /\ s e. Q. ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) )
29	26, 27, 28	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) )
30		addcomnqg 7712	. . . . . . . 8 |- ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. /\ s e. Q. ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) = ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
31	26, 27, 30	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) = ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
32	29, 31	breqtrd 4140	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33		simprr 533	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a )
34		ltsonq 7729	. . . . . . 7 |- <Q Or Q.
35		ltrelnq 7696	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
36	34, 35	sotri 5163	. . . . . 6 |- ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q a )
37	32, 33, 36	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q a )
38	10	adantr 276	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. )
39		simprl 531	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> a e. ( 1st ` ( F ` b ) ) )
40		elprnql 7812	. . . . . . 7 |- ( ( <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. /\ a e. ( 1st ` ( F ` b ) ) ) -> a e. Q. )
41	38, 39, 40	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> a e. Q. )
42		ltexnqq 7739	. . . . . 6 |- ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. /\ a e. Q. ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q a <-> E. t e. Q. ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) )
43	26, 41, 42	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) <Q a <-> E. t e. Q. ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) )
44	37, 43	mpbid 147	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> E. t e. Q. ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a )
45	27	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> s e. Q. )
46	26	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
47		addcomnqg 7712	. . . . . . . . . . 11 |- ( ( s e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) )
48	45, 46, 47	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) )
49	33	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a )
50	48, 49	eqbrtrrd 4138	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) <Q a )
51		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a )
52	50, 51	breqtrrd 4142	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) )
53		simplr 529	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> t e. Q. )
54		ltanqg 7731	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( s <Q t <-> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) ) )
55	45, 53, 46, 54	syl3anc 1274	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( s <Q t <-> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) ) )
56	52, 55	mpbird 167	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> s <Q t )
57	7	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> b e. N. )
58		addcomnqg 7712	. . . . . . . . . . . . 13 |- ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. /\ t e. Q. ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
59	46, 53, 58	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
60	59, 51	eqtr3d 2269	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = a )
61	39	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> a e. ( 1st ` ( F ` b ) ) )
62	60, 61	eqeltrd 2311	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) )
63		addclnq 7706	. . . . . . . . . . . 12 |- ( ( t e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
64	53, 46, 63	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
65	8	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( F ` b ) e. P. )
66		nqprl 7882	. . . . . . . . . . 11 |- ( ( ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. /\ ( F ` b ) e. P. ) -> ( ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) <-> <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
67	64, 65, 66	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. ( 1st ` ( F ` b ) ) <-> <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
68	62, 67	mpbid 147	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
69		opeq1 3888	. . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
70	69	eceq1d 6816	. . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
71	70	fveq2d 5679	. . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
72	71	oveq2d 6074	. . . . . . . . . . . . . 14 |- ( r = b -> ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
73	72	breq2d 4126	. . . . . . . . . . . . 13 |- ( r = b -> ( p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
74	73	abbidv 2354	. . . . . . . . . . . 12 |- ( r = b -> { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
75	72	breq1d 4124	. . . . . . . . . . . . 13 |- ( r = b -> ( ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
76	75	abbidv 2354	. . . . . . . . . . . 12 |- ( r = b -> { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } = { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } )
77	74, 76	opeq12d 3896	. . . . . . . . . . 11 |- ( r = b -> <. { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. )
78		fveq2 5675	. . . . . . . . . . 11 |- ( r = b -> ( F ` r ) = ( F ` b ) )
79	77, 78	breq12d 4127	. . . . . . . . . 10 |- ( r = b -> ( <. { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
80	79	rspcev 2923	. . . . . . . . 9 |- ( ( b e. N. /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> E. r e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
81	57, 68, 80	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> E. r e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
82	1	caucvgprprlemell 8016	. . . . . . . 8 |- ( t e. ( 1st ` L ) <-> ( t e. Q. /\ E. r e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
83	53, 81, 82	sylanbrc 417	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> t e. ( 1st ` L ) )
84	56, 83	jca 306	. . . . . 6 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) /\ ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a ) -> ( s <Q t /\ t e. ( 1st ` L ) ) )
85	84	ex 115	. . . . 5 |- ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) /\ t e. Q. ) -> ( ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a -> ( s <Q t /\ t e. ( 1st ` L ) ) ) )
86	85	reximdva 2646	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> ( E. t e. Q. ( ( *Q ` [ <. b , 1o >. ] ~Q ) +Q t ) = a -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) )
87	44, 86	mpd 13	. . 3 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) /\ ( a e. ( 1st ` ( F ` b ) ) /\ ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q a ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )
88	25, 87	rexlimddv 2667	. 2 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( b e. N. /\ <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )
89	4, 88	rexlimddv 2667	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   1oc1o 6653   [cec 6778   N.cnpi 7603   <N clti 7606   ~Q ceq 7610   Q.cnq 7611   +Q cplq 7613   *Qcrq 7615   <Q cltq 7616   P.cnp 7622   +P. cpp 7624   <P cltp 7626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-inp 7797  df-iltp 7801

This theorem is referenced by:  caucvgprprlemrnd  8032