Theorem cauappcvgprlemopu 7979

Description: Lemma for cauappcvgpr 7993. 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 4118	. . . . . 6 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
2	1	rexbidv 2545	. . . . 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 5678	. . . . . 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 7694	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4261	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
7	5	rabex 4261	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
8	6, 7	op2nd 6354	. . . . . 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 2255	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
10	2, 9	elrab2 2979	. . . 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 7746	. . . 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 2593	. . . . . . . 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 4118	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
24	23	rexbidv 2545	. . . . . . . 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 2979	. . . . . . 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 2646	. . 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 2667	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 2205   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  cauappcvgprlemrnd  7981