Theorem cauappcvgprlemupu 7859

Description: Lemma for cauappcvgpr 7872. The upper cut of the putative limit is upper. (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
cauappcvgprlemupu	|- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r 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 cauappcvgprlemupu

Step	Hyp	Ref	Expression
1		ltrelnq 7575	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4776	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simprd 114	. . 3 |- ( s <Q r -> r e. Q. )
4	3	3ad2ant2 1043	. 2 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. Q. )
5		breq2 4090	. . . . . . 7 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
6	5	rexbidv 2531	. . . . . 6 |- ( u = s -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
7		cauappcvgpr.lim	. . . . . . . 8 |- 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 } >.
8	7	fveq2i 5638	. . . . . . 7 |- ( 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 } >. )
9		nqex 7573	. . . . . . . . 9 |- Q. e. _V
10	9	rabex 4232	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
11	9	rabex 4232	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
12	10, 11	op2nd 6305	. . . . . . 7 |- ( 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 }
13	8, 12	eqtri 2250	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
14	6, 13	elrab2 2963	. . . . 5 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
15	14	simprbi 275	. . . 4 |- ( s e. ( 2nd ` L ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
16	15	3ad2ant3 1044	. . 3 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
17		ltsonq 7608	. . . . . . 7 |- <Q Or Q.
18	17, 1	sotri 5130	. . . . . 6 |- ( ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) -> ( ( F ` q ) +Q q ) <Q r )
19	18	expcom 116	. . . . 5 |- ( s <Q r -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( F ` q ) +Q q ) <Q r ) )
20	19	3ad2ant2 1043	. . . 4 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( F ` q ) +Q q ) <Q r ) )
21	20	reximdv 2631	. . 3 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q s -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
22	16, 21	mpd 13	. 2 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r )
23		breq2 4090	. . . 4 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
24	23	rexbidv 2531	. . 3 |- ( u = r -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
25	24, 13	elrab2 2963	. 2 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
26	4, 22, 25	sylanbrc 417	1 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3670   class class class wbr 4086   -->wf 5320   ` cfv 5324  (class class class)co 6013   2ndc2nd 6297   Q.cnq 7490   +Q cplq 7492   <Q cltq 7495

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-enq 7557  df-nqqs 7558  df-ltnqqs 7563

This theorem is referenced by:  cauappcvgprlemrnd  7860