Theorem caucvgprlemopl 7736

Description: Lemma for caucvgpr 7749. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.

Assertion

Ref	Expression
caucvgprlemopl	|- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )

Distinct variable groups:   A, j   F, l, r, s   u, F   j, L, r, s   j, l, s   ph, j, r, s   u, j, r, s

Allowed substitution hints:   ph( u, k, n, l)   A( u, k, n, s, r, l)   F( j, k, n)   L( u, k, n, l)

Proof of Theorem caucvgprlemopl

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5929	. . . . . . 7 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 4043	. . . . . 6 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2498	. . . . 5 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4		caucvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
5	4	fveq2i 5561	. . . . . 6 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
6		nqex 7430	. . . . . . . 8 |- Q. e. _V
7	6	rabex 4177	. . . . . . 7 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 4177	. . . . . . 7 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 6204	. . . . . 6 |- ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
10	5, 9	eqtri 2217	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2923	. . . 4 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
12	11	simprbi 275	. . 3 |- ( s e. ( 1st ` L ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
13	12	adantl 277	. 2 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
14		simprr 531	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
15		ltbtwnnqq 7482	. . . 4 |- ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. t e. Q. ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) )
16	14, 15	sylib 122	. . 3 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) -> E. t e. Q. ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) )
17		simplrl 535	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> j e. N. )
18		nnnq 7489	. . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
19		recclnq 7459	. . . . . . . . 9 |- ( [ <. j , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
20	17, 18, 19	3syl 17	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
21	11	simplbi 274	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
22	21	ad3antlr 493	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> s e. Q. )
23		ltaddnq 7474	. . . . . . . 8 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ s e. Q. ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) )
24	20, 22, 23	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) )
25		addcomnqg 7448	. . . . . . . 8 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ s e. Q. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
26	20, 22, 25	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
27	24, 26	breqtrd 4059	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
28		simprrl 539	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t )
29		ltsonq 7465	. . . . . . 7 |- <Q Or Q.
30		ltrelnq 7432	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 5065	. . . . . 6 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q t )
32	27, 28, 31	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q t )
33		simprl 529	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> t e. Q. )
34		ltexnqq 7475	. . . . . 6 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ t e. Q. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q t <-> E. r e. Q. ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) )
35	20, 33, 34	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q t <-> E. r e. Q. ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) )
36	32, 35	mpbid 147	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> E. r e. Q. ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t )
37	22	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> s e. Q. )
38	20	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
39		addcomnqg 7448	. . . . . . . . . . 11 |- ( ( s e. Q. /\ ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) )
40	37, 38, 39	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) )
41	28	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t )
42	40, 41	eqbrtrrd 4057	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) <Q t )
43		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t )
44	42, 43	breqtrrd 4061	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) )
45		simplr 528	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> r e. Q. )
46		ltanqg 7467	. . . . . . . . 9 |- ( ( s e. Q. /\ r e. Q. /\ ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. ) -> ( s <Q r <-> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) ) )
47	37, 45, 38, 46	syl3anc 1249	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( s <Q r <-> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q s ) <Q ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) ) )
48	44, 47	mpbird 167	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> s <Q r )
49	17	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> j e. N. )
50		simprrr 540	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> t <Q ( F ` j ) )
51	50	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> t <Q ( F ` j ) )
52		addcomnqg 7448	. . . . . . . . . . . . 13 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ r e. Q. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
53	38, 45, 52	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
54	53, 43	eqtr3d 2231	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = t )
55	54	breq1d 4043	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> t <Q ( F ` j ) ) )
56	51, 55	mpbird 167	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
57		rspe 2546	. . . . . . . . 9 |- ( ( j e. N. /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
58	49, 56, 57	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
59		oveq1 5929	. . . . . . . . . . 11 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
60	59	breq1d 4043	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
61	60	rexbidv 2498	. . . . . . . . 9 |- ( l = r -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
62	61, 10	elrab2 2923	. . . . . . . 8 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
63	45, 58, 62	sylanbrc 417	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> r e. ( 1st ` L ) )
64	48, 63	jca 306	. . . . . 6 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) /\ ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t ) -> ( s <Q r /\ r e. ( 1st ` L ) ) )
65	64	ex 115	. . . . 5 |- ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) /\ r e. Q. ) -> ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t -> ( s <Q r /\ r e. ( 1st ` L ) ) ) )
66	65	reximdva 2599	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> ( E. r e. Q. ( ( *Q ` [ <. j , 1o >. ] ~Q ) +Q r ) = t -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) )
67	36, 66	mpd 13	. . 3 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) /\ ( t e. Q. /\ ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q t /\ t <Q ( F ` j ) ) ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
68	16, 67	rexlimddv 2619	. 2 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( j e. N. /\ ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
69	13, 68	rexlimddv 2619	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   <.cop 3625   class class class wbr 4033   -->wf 5254   ` cfv 5258  (class class class)co 5922   1stc1st 6196   1oc1o 6467   [cec 6590   N.cnpi 7339   <N clti 7342   ~Q ceq 7346   Q.cnq 7347   +Q cplq 7349   *Qcrq 7351   <Q cltq 7352

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420

This theorem is referenced by:  caucvgprlemrnd  7740