Theorem cauappcvgprlemupu 7868

Description: Lemma for cauappcvgpr 7881. 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 7584	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4778	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simprd 114	. . 3 |- ( s <Q r -> r e. Q. )
4	3	3ad2ant2 1045	. 2 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. Q. )
5		breq2 4092	. . . . . . 7 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
6	5	rexbidv 2533	. . . . . 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 5642	. . . . . . 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 7582	. . . . . . . . 9 |- Q. e. _V
10	9	rabex 4234	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
11	9	rabex 4234	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
12	10, 11	op2nd 6309	. . . . . . 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 2252	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
14	6, 13	elrab2 2965	. . . . 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 1046	. . 3 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
17		ltsonq 7617	. . . . . . 7 |- <Q Or Q.
18	17, 1	sotri 5132	. . . . . 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 1045	. . . 4 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( F ` q ) +Q q ) <Q r ) )
21	20	reximdv 2633	. . 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 4092	. . . 4 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
24	23	rexbidv 2533	. . 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 2965	. 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   {crab 2514   <.cop 3672   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6017   2ndc2nd 6301   Q.cnq 7499   +Q cplq 7501   <Q cltq 7504

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-mi 7525  df-lti 7526  df-enq 7566  df-nqqs 7567  df-ltnqqs 7572

This theorem is referenced by:  cauappcvgprlemrnd  7869