Theorem cauappcvgprlemlol 7598

Description: Lemma for cauappcvgpr 7613. 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 7316	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4661	. . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
3	2	simpld 111	. . 3 |- ( s <Q r -> s e. Q. )
4	3	3ad2ant2 1014	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5		oveq1 5858	. . . . . . . 8 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
6	5	breq1d 3997	. . . . . . 7 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
7	6	rexbidv 2471	. . . . . 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 5497	. . . . . . 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 7314	. . . . . . . . 9 |- Q. e. _V
11	10	rabex 4131	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
12	10	rabex 4131	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
13	11, 12	op1st 6123	. . . . . . 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 2191	. . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
15	7, 14	elrab2 2889	. . . . 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 1015	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
18		simpll2 1032	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> s <Q r )
19		ltanqg 7351	. . . . . . . . 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 485	. . . . . . . 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 1014	. . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
24	23	ad2antrr 485	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> r e. Q. )
25		simplr 525	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ q e. Q. ) /\ ( r +Q q ) <Q ( F ` q ) ) -> q e. Q. )
26		addcomnqg 7332	. . . . . . . . 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 6020	. . . . . . 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 7349	. . . . . . 7 |- <Q Or Q.
31	30, 1	sotri 5004	. . . . . 6 |- ( ( ( s +Q q ) <Q ( r +Q q ) /\ ( r +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
32	29, 31	sylancom 418	. . . . 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 2572	. . 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 5858	. . . . 5 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
37	36	breq1d 3997	. . . 4 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
38	37	rexbidv 2471	. . 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 2889	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
40	4, 35, 39	sylanbrc 415	1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3584   class class class wbr 3987   -->wf 5192   ` cfv 5196  (class class class)co 5851   1stc1st 6115   Q.cnq 7231   +Q cplq 7233   <Q cltq 7236

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-oadd 6397  df-omul 6398  df-er 6510  df-ec 6512  df-qs 6516  df-ni 7255  df-pli 7256  df-mi 7257  df-lti 7258  df-plpq 7295  df-enq 7298  df-nqqs 7299  df-plqqs 7300  df-ltnqqs 7304

This theorem is referenced by:  cauappcvgprlemrnd  7601