Theorem caucvgprlemopu 7669

Description: Lemma for caucvgpr 7680. The upper 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
caucvgprlemopu	|- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` 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 caucvgprlemopu

Step	Hyp	Ref	Expression
1		breq2 4007	. . . . . 6 |- ( u = r -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
2	1	rexbidv 2478	. . . . 5 |- ( u = r -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
3		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 } >.
4	3	fveq2i 5518	. . . . . 6 |- ( 2nd ` L ) = ( 2nd ` <. { 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		nqex 7361	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4147	. . . . . . 7 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
7	5	rabex 4147	. . . . . . 7 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
8	6, 7	op2nd 6147	. . . . . 6 |- ( 2nd ` <. { 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 } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
9	4, 8	eqtri 2198	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
10	2, 9	elrab2 2896	. . . 4 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
11	10	simprbi 275	. . 3 |- ( r e. ( 2nd ` L ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r )
12	11	adantl 277	. 2 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r )
13		simprr 531	. . . 4 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r )
14		ltbtwnnqq 7413	. . . 4 |- ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r <-> E. s e. Q. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) )
15	13, 14	sylib 122	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) -> E. s e. Q. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) )
16		simprr 531	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> s <Q r )
17		simplr 528	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> s e. Q. )
18		simplrl 535	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) -> j e. N. )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> j e. N. )
20		simprl 529	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
21		rspe 2526	. . . . . . . 8 |- ( ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
22	19, 20, 21	syl2anc 411	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
23		breq2 4007	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
24	23	rexbidv 2478	. . . . . . . 8 |- ( u = s -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
25	24, 9	elrab2 2896	. . . . . . 7 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
26	17, 22, 25	sylanbrc 417	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> s e. ( 2nd ` L ) )
27	16, 26	jca 306	. . . . 5 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) )
28	27	ex 115	. . . 4 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) /\ s e. Q. ) -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
29	28	reximdva 2579	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) -> ( E. s e. Q. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s /\ s <Q r ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
30	15, 29	mpd 13	. 2 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( j e. N. /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
31	12, 30	rexlimddv 2599	1 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3595   class class class wbr 4003   -->wf 5212   ` cfv 5216  (class class class)co 5874   2ndc2nd 6139   1oc1o 6409   [cec 6532   N.cnpi 7270   <N clti 7273   ~Q ceq 7277   Q.cnq 7278   +Q cplq 7280   *Qcrq 7282   <Q cltq 7283

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351

This theorem is referenced by:  caucvgprlemrnd  7671