Theorem cauappcvgprlemupu 7832

Description: Lemma for cauappcvgpr 7845. 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 7548	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4770	. . . 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 4086	. . . . . . 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 5629	. . . . . . 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 7546	. . . . . . . . 9 |- Q. e. _V
10	9	rabex 4227	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
11	9	rabex 4227	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
12	10, 11	op2nd 6291	. . . . . . 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 2962	. . . . 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 7581	. . . . . . 7 |- <Q Or Q.
18	17, 1	sotri 5123	. . . . . 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 4086	. . . 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 2962	. 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 3669   class class class wbr 4082   -->wf 5313   ` cfv 5317  (class class class)co 6000   2ndc2nd 6283   Q.cnq 7463   +Q cplq 7465   <Q cltq 7468

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-mi 7489  df-lti 7490  df-enq 7530  df-nqqs 7531  df-ltnqqs 7536

This theorem is referenced by:  cauappcvgprlemrnd  7833