Theorem cauappcvgprlemopu 7911

Description: Lemma for cauappcvgpr 7925. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )
cauappcvgpr.lim	|- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.

Assertion

Ref	Expression
cauappcvgprlemopu	|- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

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

Allowed substitution hints:   ph( u, l)   A( u, r, q, l)   L( u, l)

Proof of Theorem cauappcvgprlemopu

Step	Hyp	Ref	Expression
1		breq2 4097	. . . . . 6 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
2	1	rexbidv 2534	. . . . 5 |- ( u = r -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
3		cauappcvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
4	3	fveq2i 5651	. . . . . 6 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. )
5		nqex 7626	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4239	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
7	5	rabex 4239	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
8	6, 7	op2nd 6319	. . . . . 6 |- ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
9	4, 8	eqtri 2252	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
10	2, 9	elrab2 2966	. . . 4 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
11	10	simprbi 275	. . 3 |- ( r e. ( 2nd ` L ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r )
12	11	adantl 277	. 2 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r )
13		simprr 533	. . . 4 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> ( ( F ` q ) +Q q ) <Q r )
14		ltbtwnnqq 7678	. . . 4 |- ( ( ( F ` q ) +Q q ) <Q r <-> E. s e. Q. ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) )
15	13, 14	sylib 122	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> E. s e. Q. ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) )
16		simprr 533	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s <Q r )
17		simplr 529	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s e. Q. )
18		simplrl 537	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) -> q e. Q. )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> q e. Q. )
20		simprl 531	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> ( ( F ` q ) +Q q ) <Q s )
21		rspe 2582	. . . . . . . 8 |- ( ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q s ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
22	19, 20, 21	syl2anc 411	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
23		breq2 4097	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
24	23	rexbidv 2534	. . . . . . . 8 |- ( u = s -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
25	24, 9	elrab2 2966	. . . . . . 7 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
26	17, 22, 25	sylanbrc 417	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s e. ( 2nd ` L ) )
27	16, 26	jca 306	. . . . 5 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) )
28	27	ex 115	. . . 4 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) -> ( ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
29	28	reximdva 2635	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> ( E. s e. Q. ( ( ( F ` q ) +Q 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 ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
31	12, 30	rexlimddv 2656	1 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   2ndc2nd 6311   Q.cnq 7543   +Q cplq 7545   <Q cltq 7548

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616

This theorem is referenced by:  cauappcvgprlemrnd  7913