Theorem caucvgprprlemopl 7529

Description: Lemma for caucvgprpr 7544. 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 7517	. . . 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 273	. . 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 275	. 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 480	. . . . . 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 521	. . . . . 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	ffvelrnd 5564	. . . . 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 7307	. . . . 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 522	. . . . 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 7517	. . . . . . . . 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 272	. . . . . . . 8 |- ( s e. ( 1st ` L ) -> s e. Q. )
14	13	ad2antlr 481	. . . . . . 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 7254	. . . . . . . . 9 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
16		recclnq 7224	. . . . . . . . 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 482	. . . . . . 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 7207	. . . . . . 7 |- ( ( s e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
20	14, 18, 19	syl2anc 409	. . . . . 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 7383	. . . . . 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 409	. . . . 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 166	. . . 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 7320	. . . 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 409	. . 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 274	. . . . . . . 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 274	. . . . . . . 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 7239	. . . . . . . 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 409	. . . . . . 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 7213	. . . . . . . 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 409	. . . . . . 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 3962	. . . . . 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 522	. . . . . 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 7230	. . . . . . 7 |- <Q Or Q.
35		ltrelnq 7197	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
36	34, 35	sotri 4942	. . . . . 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 409	. . . . 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 274	. . . . . . 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 521	. . . . . . 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 7313	. . . . . . 7 |- ( ( <. ( 1st ` ( F ` b ) ) , ( 2nd ` ( F ` b ) ) >. e. P. /\ a e. ( 1st ` ( F ` b ) ) ) -> a e. Q. )
41	38, 39, 40	syl2anc 409	. . . . . 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 7240	. . . . . 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 409	. . . . 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 146	. . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 7213	. . . . . . . . . . 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 409	. . . . . . . . . 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 480	. . . . . . . . . 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 3960	. . . . . . . . 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 109	. . . . . . . . 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 3964	. . . . . . . 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 520	. . . . . . . . 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 7232	. . . . . . . . 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 1217	. . . . . . . 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 166	. . . . . . 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 484	. . . . . . . . 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 7213	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 2175	. . . . . . . . . . 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 480	. . . . . . . . . . 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 2217	. . . . . . . . . 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 7207	. . . . . . . . . . . 12 |- ( ( t e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
64	53, 46, 63	syl2anc 409	. . . . . . . . . . 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 484	. . . . . . . . . . 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 7383	. . . . . . . . . . 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 409	. . . . . . . . . 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 146	. . . . . . . . 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 3713	. . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
70	69	eceq1d 6473	. . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
71	70	fveq2d 5433	. . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
72	71	oveq2d 5798	. . . . . . . . . . . . . 14 |- ( r = b -> ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
73	72	breq2d 3949	. . . . . . . . . . . . 13 |- ( r = b -> ( p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
74	73	abbidv 2258	. . . . . . . . . . . 12 |- ( r = b -> { p | p <Q ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
75	72	breq1d 3947	. . . . . . . . . . . . 13 |- ( r = b -> ( ( t +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
76	75	abbidv 2258	. . . . . . . . . . . 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 3721	. . . . . . . . . . 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 5429	. . . . . . . . . . 11 |- ( r = b -> ( F ` r ) = ( F ` b ) )
79	77, 78	breq12d 3950	. . . . . . . . . 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 2793	. . . . . . . . 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 409	. . . . . . . 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 7517	. . . . . . . 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 414	. . . . . . 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 304	. . . . . 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 114	. . . . 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 2537	. . . 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 2557	. 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 2557	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   1oc1o 6314   [cec 6435   N.cnpi 7104   <N clti 7107   ~Q ceq 7111   Q.cnq 7112   +Q cplq 7114   *Qcrq 7116   <Q cltq 7117   P.cnp 7123   +P. cpp 7125   <P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298  df-iltp 7302

This theorem is referenced by:  caucvgprprlemrnd  7533