Theorem caucvgprlemopl 7670

Description: Lemma for caucvgpr 7683. 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 5884	. . . . . . 7 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 4015	. . . . . 6 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2478	. . . . 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 5520	. . . . . 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 7364	. . . . . . . 8 |- Q. e. _V
7	6	rabex 4149	. . . . . . 7 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 4149	. . . . . . 7 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 6149	. . . . . 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 2198	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2898	. . . 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 7416	. . . 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 7423	. . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
19		recclnq 7393	. . . . . . . . 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 7408	. . . . . . . 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 7382	. . . . . . . 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 4031	. . . . . 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 7399	. . . . . . 7 |- <Q Or Q.
30		ltrelnq 7366	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 5026	. . . . . 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 7409	. . . . . 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 7382	. . . . . . . . . . 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 4029	. . . . . . . . 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 4033	. . . . . . . 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 7401	. . . . . . . . 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 1238	. . . . . . . 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 7382	. . . . . . . . . . . . 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 2212	. . . . . . . . . . 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 4015	. . . . . . . . . 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 2526	. . . . . . . . 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 5884	. . . . . . . . . . 11 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
60	59	breq1d 4015	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
61	60	rexbidv 2478	. . . . . . . . 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 2898	. . . . . . . 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 2579	. . . 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 2599	. 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 2599	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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   Q.cnq 7281   +Q cplq 7283   *Qcrq 7285   <Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354

This theorem is referenced by:  caucvgprlemrnd  7674