Theorem cauappcvgprlemlol 6745

Description: Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (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
cauappcvgprlemlol	|- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` 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 cauappcvgprlemlol

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 6463	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4392	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simpld 105	. . 3 |- ( s <Q r -> s e. Q. )
4	3	3ad2ant2 926	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5		oveq1 5519	. . . . . . . 8 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
6	5	breq1d 3774	. . . . . . 7 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
7	6	rexbidv 2327	. . . . . 6 |- ( l = r -> ( E. q e. Q. ( l +Q q ) <Q ( F ` q ) <-> E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
8		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 } >.
9	8	fveq2i 5181	. . . . . . 7 |- ( 1st ` L ) = ( 1st ` <. { 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 } >. )
10		nqex 6461	. . . . . . . . 9 |- Q. e. _V
11	10	rabex 3901	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
12	10	rabex 3901	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
13	11, 12	op1st 5773	. . . . . . 7 |- ( 1st ` <. { 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 } >. ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
14	9, 13	eqtri 2060	. . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
15	7, 14	elrab2 2700	. . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
16	15	simprbi 260	. . . 4 |- ( r e. ( 1st ` L ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
17	16	3ad2ant3 927	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
18		simpll2 944	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s <Q r )
19		ltanqg 6498	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
20	19	adantl 262	. . . . . . . 8 |- ( ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
21	4	ad2antrr 457	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s e. Q. )
22	2	simprd 107	. . . . . . . . . 10 |- ( s <Q r -> r e. Q. )
23	22	3ad2ant2 926	. . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
24	23	ad2antrr 457	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> r e. Q. )
25		simplr 482	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> q e. Q. )
26		addcomnqg 6479	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
27	26	adantl 262	. . . . . . . 8 |- ( ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
28	20, 21, 24, 25, 27	caovord2d 5670	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s <Q r <-> ( s +Q q ) <Q ( r +Q q ) ) )
29	18, 28	mpbid 135	. . . . . 6 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( r +Q q ) )
30		ltsonq 6496	. . . . . . 7 |- <Q Or Q.
31	30, 1	sotri 4720	. . . . . 6 |- ( ( ( s +Q q ) <Q ( r +Q q ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
32	29, 31	sylancom 397	. . . . 5 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
33	32	ex 108	. . . 4 |- ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) -> ( ( r +Q q ) <Q ( F ` q ) -> ( s +Q q ) <Q ( F ` q ) ) )
34	33	reximdva 2421	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> ( E. q e. Q. ( r +Q q ) <Q ( F ` q ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
35	17, 34	mpd 13	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
36		oveq1 5519	. . . . 5 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
37	36	breq1d 3774	. . . 4 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
38	37	rexbidv 2327	. . 3 |- ( l = s -> ( E. q e. Q. ( l +Q q ) <Q ( F ` q ) <-> E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
39	38, 14	elrab2 2700	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
40	4, 35, 39	sylanbrc 394	1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   Q.cnq 6378   +Q cplq 6380   <Q cltq 6383

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-ltnqqs 6451

This theorem is referenced by:  cauappcvgprlemrnd  6748