Theorem caucvgprlemopu 7858

Description: Lemma for caucvgpr 7869. 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 4087	. . . . . 6 |- ( u = r -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
2	1	rexbidv 2531	. . . . 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 5630	. . . . . 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 7550	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4228	. . . . . . 7 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
7	5	rabex 4228	. . . . . . 7 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
8	6, 7	op2nd 6293	. . . . . 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 2250	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
10	2, 9	elrab2 2962	. . . 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 7602	. . . 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 2579	. . . . . . . 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 4087	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
24	23	rexbidv 2531	. . . . . . . 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 2962	. . . . . . 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 2632	. . 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 2653	1 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4083   -->wf 5314   ` cfv 5318  (class class class)co 6001   2ndc2nd 6285   1oc1o 6555   [cec 6678   N.cnpi 7459   <N clti 7462   ~Q ceq 7466   Q.cnq 7467   +Q cplq 7469   *Qcrq 7471   <Q cltq 7472

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540

This theorem is referenced by:  caucvgprlemrnd  7860