Theorem cauappcvgprlemlol 7964

Description: Lemma for cauappcvgpr 7979. 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 7682	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4804	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simpld 112	. . 3 |- ( s <Q r -> s e. Q. )
4	3	3ad2ant2 1046	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5		oveq1 6059	. . . . . . . 8 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
6	5	breq1d 4121	. . . . . . 7 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
7	6	rexbidv 2545	. . . . . 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 5675	. . . . . . 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 7680	. . . . . . . . 9 |- Q. e. _V
11	10	rabex 4258	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
12	10	rabex 4258	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
13	11, 12	op1st 6342	. . . . . . 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 2255	. . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
15	7, 14	elrab2 2978	. . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
16	15	simprbi 275	. . . 4 |- ( r e. ( 1st ` L ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
17	16	3ad2ant3 1047	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
18		simpll2 1064	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s <Q r )
19		ltanqg 7717	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
20	19	adantl 277	. . . . . . . 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 488	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s e. Q. )
22	2	simprd 114	. . . . . . . . . 10 |- ( s <Q r -> r e. Q. )
23	22	3ad2ant2 1046	. . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
24	23	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> r e. Q. )
25		simplr 529	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> q e. Q. )
26		addcomnqg 7698	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
27	26	adantl 277	. . . . . . . 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 6226	. . . . . . 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 147	. . . . . 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 7715	. . . . . . 7 |- <Q Or Q.
31	30, 1	sotri 5160	. . . . . 6 |- ( ( ( s +Q q ) <Q ( r +Q q ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
32	29, 31	sylancom 420	. . . . 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 115	. . . 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 2646	. . 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 6059	. . . . 5 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
37	36	breq1d 4121	. . . 4 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
38	37	rexbidv 2545	. . 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 2978	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
40	4, 35, 39	sylanbrc 417	1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3694   class class class wbr 4111   -->wf 5350   ` cfv 5354  (class class class)co 6052   1stc1st 6334   Q.cnq 7597   +Q cplq 7599   <Q cltq 7602

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-ltnqqs 7670

This theorem is referenced by:  cauappcvgprlemrnd  7967