Theorem caucvgprprlemlol 7499

Description: Lemma for caucvgprpr 7513. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   A, m   m, F   F, l   u, F, r   p, l, s   q, l, s, r   t, l, p   u, q, s, r   u, p, t, r   ph, r   r, q, t

Allowed substitution hints:   ph( u, t, k, m, n, s, q, p, l)   A( u, t, k, n, s, r, q, p, l)   F( t, k, n, s, q, p)   L( u, t, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemlol

Dummy variables b x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 7166	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4586	. . . 4 |- ( s <Q t -> ( s e. Q. /\ t e. Q. ) )
3	2	simpld 111	. . 3 |- ( s <Q t -> s e. Q. )
4	3	3ad2ant2 1003	. 2 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. Q. )
5		caucvgprpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
6	5	caucvgprprlemell 7486	. . . . . 6 |- ( t e. ( 1st ` L ) <-> ( t e. Q. /\ E. b e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
7	6	simprbi 273	. . . . 5 |- ( t e. ( 1st ` L ) -> E. b e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
8	7	3ad2ant3 1004	. . . 4 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> E. b e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
9		simpll2 1021	. . . . . . . . 9 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> s <Q t )
10		ltanqg 7201	. . . . . . . . . . 11 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
11	10	adantl 275	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) /\ ( x e. Q. /\ y e. Q. /\ z e. Q. ) ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
12	4	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> s e. Q. )
13	2	simprd 113	. . . . . . . . . . . 12 |- ( s <Q t -> t e. Q. )
14	13	3ad2ant2 1003	. . . . . . . . . . 11 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> t e. Q. )
15	14	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> t e. Q. )
16		simplr 519	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> b e. N. )
17		nnnq 7223	. . . . . . . . . . 11 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
18		recclnq 7193	. . . . . . . . . . 11 |- ( [ <. b , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
19	16, 17, 18	3syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
20		addcomnqg 7182	. . . . . . . . . . 11 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
21	20	adantl 275	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) /\ ( x e. Q. /\ y e. Q. ) ) -> ( x +Q y ) = ( y +Q x ) )
22	11, 12, 15, 19, 21	caovord2d 5933	. . . . . . . . 9 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> ( s <Q t <-> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
23	9, 22	mpbid 146	. . . . . . . 8 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
24		ltnqpri 7395	. . . . . . . 8 |- ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. )
25	23, 24	syl 14	. . . . . . 7 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. )
26		ltsopr 7397	. . . . . . . 8 |- <P Or P.
27		ltrelpr 7306	. . . . . . . 8 |- <P C_ ( P. X. P. )
28	26, 27	sotri 4929	. . . . . . 7 |- ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
29	25, 28	sylancom 416	. . . . . 6 |- ( ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) /\ <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
30	29	ex 114	. . . . 5 |- ( ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) /\ b e. N. ) -> ( <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
31	30	reximdva 2532	. . . 4 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> ( E. b e. N. <. { p | p <Q ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( t +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) -> E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
32	8, 31	mpd 13	. . 3 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
33		opeq1 3700	. . . . . . . . . . 11 |- ( b = r -> <. b , 1o >. = <. r , 1o >. )
34	33	eceq1d 6458	. . . . . . . . . 10 |- ( b = r -> [ <. b , 1o >. ] ~Q = [ <. r , 1o >. ] ~Q )
35	34	fveq2d 5418	. . . . . . . . 9 |- ( b = r -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. r , 1o >. ] ~Q ) )
36	35	oveq2d 5783	. . . . . . . 8 |- ( b = r -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	breq2d 3936	. . . . . . 7 |- ( b = r -> ( p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
38	37	abbidv 2255	. . . . . 6 |- ( b = r -> { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
39	36	breq1d 3934	. . . . . . 7 |- ( b = r -> ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
40	39	abbidv 2255	. . . . . 6 |- ( b = r -> { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } = { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } )
41	38, 40	opeq12d 3708	. . . . 5 |- ( b = r -> <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. )
42		fveq2 5414	. . . . 5 |- ( b = r -> ( F ` b ) = ( F ` r ) )
43	41, 42	breq12d 3937	. . . 4 |- ( b = r -> ( <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
44	43	cbvrexv 2653	. . 3 |- ( E. b e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) <-> E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
45	32, 44	sylib 121	. 2 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
46	5	caucvgprprlemell 7486	. 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
47	4, 45, 46	sylanbrc 413	1 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   Q.cnq 7081   +Q cplq 7083   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   +P. cpp 7094   <P cltp 7096

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271

This theorem is referenced by:  caucvgprprlemrnd  7502