Theorem cauappcvgprlemopu 7638

Description: Lemma for cauappcvgpr 7652. 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 4004	. . . . . 6 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
2	1	rexbidv 2478	. . . . 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 5514	. . . . . 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 7353	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4144	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
7	5	rabex 4144	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
8	6, 7	op2nd 6142	. . . . . 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 2198	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
10	2, 9	elrab2 2896	. . . 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 531	. . . 4 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> ( ( F ` q ) +Q q ) <Q r )
14		ltbtwnnqq 7405	. . . 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 531	. . . . . 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 528	. . . . . . 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 535	. . . . . . . . 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 529	. . . . . . . 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 2526	. . . . . . . 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 4004	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
24	23	rexbidv 2478	. . . . . . . 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 2896	. . . . . . 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 2579	. . 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 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 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   2ndc2nd 6134   Q.cnq 7270   +Q cplq 7272   <Q cltq 7275

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343

This theorem is referenced by:  cauappcvgprlemrnd  7640