Theorem cauappcvgprlemlol 7479

Description: Lemma for cauappcvgpr 7494. 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 7197	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4599	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simpld 111	. . 3 |- ( s <Q r -> s e. Q. )
4	3	3ad2ant2 1004	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5		oveq1 5789	. . . . . . . 8 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
6	5	breq1d 3947	. . . . . . 7 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
7	6	rexbidv 2439	. . . . . 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 5432	. . . . . . 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 7195	. . . . . . . . 9 |- Q. e. _V
11	10	rabex 4080	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
12	10	rabex 4080	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
13	11, 12	op1st 6052	. . . . . . 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 2161	. . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
15	7, 14	elrab2 2847	. . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
16	15	simprbi 273	. . . 4 |- ( r e. ( 1st ` L ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
17	16	3ad2ant3 1005	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
18		simpll2 1022	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s <Q r )
19		ltanqg 7232	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
20	19	adantl 275	. . . . . . . 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 480	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s e. Q. )
22	2	simprd 113	. . . . . . . . . 10 |- ( s <Q r -> r e. Q. )
23	22	3ad2ant2 1004	. . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
24	23	ad2antrr 480	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> r e. Q. )
25		simplr 520	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> q e. Q. )
26		addcomnqg 7213	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
27	26	adantl 275	. . . . . . . 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 5948	. . . . . . 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 146	. . . . . 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 7230	. . . . . . 7 |- <Q Or Q.
31	30, 1	sotri 4942	. . . . . 6 |- ( ( ( s +Q q ) <Q ( r +Q q ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
32	29, 31	sylancom 417	. . . . 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 114	. . . 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 2537	. . 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 5789	. . . . 5 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
37	36	breq1d 3947	. . . 4 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
38	37	rexbidv 2439	. . 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 2847	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
40	4, 35, 39	sylanbrc 414	1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-ltnqqs 7185

This theorem is referenced by:  cauappcvgprlemrnd  7482