Theorem cauappcvgprlemupu 7457

Description: Lemma for cauappcvgpr 7470. 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 7173	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4591	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simprd 113	. . 3 |- ( s <Q r -> r e. Q. )
4	3	3ad2ant2 1003	. 2 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. Q. )
5		breq2 3933	. . . . . . 7 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
6	5	rexbidv 2438	. . . . . 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 5424	. . . . . . 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 7171	. . . . . . . . 9 |- Q. e. _V
10	9	rabex 4072	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
11	9	rabex 4072	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
12	10, 11	op2nd 6045	. . . . . . 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 2160	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
14	6, 13	elrab2 2843	. . . . 5 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
15	14	simprbi 273	. . . 4 |- ( s e. ( 2nd ` L ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
16	15	3ad2ant3 1004	. . 3 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
17		ltsonq 7206	. . . . . . 7 |- <Q Or Q.
18	17, 1	sotri 4934	. . . . . 6 |- ( ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) -> ( ( F ` q ) +Q q ) <Q r )
19	18	expcom 115	. . . . 5 |- ( s <Q r -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( F ` q ) +Q q ) <Q r ) )
20	19	3ad2ant2 1003	. . . 4 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( F ` q ) +Q q ) <Q r ) )
21	20	reximdv 2533	. . 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 3933	. . . 4 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
24	23	rexbidv 2438	. . 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 2843	. 2 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
26	4, 22, 25	sylanbrc 413	1 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   2ndc2nd 6037   Q.cnq 7088   +Q cplq 7090   <Q cltq 7093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-mi 7114  df-lti 7115  df-enq 7155  df-nqqs 7156  df-ltnqqs 7161

This theorem is referenced by:  cauappcvgprlemrnd  7458