Theorem caucvgprlemlol 7632

Description: Lemma for caucvgpr 7644. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.

Assertion

Ref	Expression
caucvgprlemlol	|- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Distinct variable groups:   A, j   F, l, r, s   u, F   j, L, r, s   j, l, s   ph, j, r, s   u, j, r, s

Allowed substitution hints:   ph( u, k, n, l)   A( u, k, n, s, r, l)   F( j, k, n)   L( u, k, n, l)

Proof of Theorem caucvgprlemlol

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

Step	Hyp	Ref	Expression
1		ltrelnq 7327	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4663	. . . 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 5860	. . . . . . . 8 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
6	5	breq1d 3999	. . . . . . 7 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
7	6	rexbidv 2471	. . . . . 6 |- ( l = r -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
8		caucvgpr.lim	. . . . . . . 8 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
9	8	fveq2i 5499	. . . . . . 7 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
10		nqex 7325	. . . . . . . . 9 |- Q. e. _V
11	10	rabex 4133	. . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
12	10	rabex 4133	. . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
13	11, 12	op1st 6125	. . . . . . 7 |- ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
14	9, 13	eqtri 2191	. . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
15	7, 14	elrab2 2889	. . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
16	15	simprbi 273	. . . 4 |- ( r e. ( 1st ` L ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
17	16	3ad2ant3 1015	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
18		simpll2 1032	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> s <Q r )
19		ltanqg 7362	. . . . . . . . 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 ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) /\ ( 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 ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> 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 ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> r e. Q. )
25		simplr 525	. . . . . . . . 9 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> j e. N. )
26		nnnq 7384	. . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
27		recclnq 7354	. . . . . . . . 9 |- ( [ <. j , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
28	25, 26, 27	3syl 17	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
29		addcomnqg 7343	. . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 275	. . . . . . . 8 |- ( ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
31	20, 21, 24, 28, 30	caovord2d 6022	. . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s <Q r <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) ) )
32	18, 31	mpbid 146	. . . . . 6 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
33		ltsonq 7360	. . . . . . 7 |- <Q Or Q.
34	33, 1	sotri 5006	. . . . . 6 |- ( ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
35	32, 34	sylancom 418	. . . . 5 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
36	35	ex 114	. . . 4 |- ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) -> ( ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
37	36	reximdva 2572	. . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> ( E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
38	17, 37	mpd 13	. 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
39		oveq1 5860	. . . . 5 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
40	39	breq1d 3999	. . . 4 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
41	40	rexbidv 2471	. . 3 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
42	41, 14	elrab2 2889	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
43	4, 38, 42	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 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247

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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  caucvgprlemrnd  7635