Theorem cauappcvgprlemopu 7743

Description: Lemma for cauappcvgpr 7757. 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 4047	. . . . . 6 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
2	1	rexbidv 2506	. . . . 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 5573	. . . . . 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 7458	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4187	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
7	5	rabex 4187	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
8	6, 7	op2nd 6223	. . . . . 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 2225	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
10	2, 9	elrab2 2931	. . . 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 7510	. . . 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 2554	. . . . . . . 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 4047	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
24	23	rexbidv 2506	. . . . . . . 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 2931	. . . . . . 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 2607	. . 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 2627	1 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   <.cop 3635   class class class wbr 4043   -->wf 5264   ` cfv 5268  (class class class)co 5934   2ndc2nd 6215   Q.cnq 7375   +Q cplq 7377   <Q cltq 7380

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-rq 7447  df-ltnqqs 7448

This theorem is referenced by:  cauappcvgprlemrnd  7745