Theorem caucvgprprlemlol 7639

Description: Lemma for caucvgprpr 7653. 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 7306	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4656	. . . 4 |- ( s <Q t -> ( s e. Q. /\ t e. Q. ) )
3	2	simpld 111	. . 3 |- ( s <Q t -> s e. Q. )
4	3	3ad2ant2 1009	. 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 7626	. . . . . 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 1010	. . . 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 1027	. . . . . . . . 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 7341	. . . . . . . . . . 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 480	. . . . . . . . . 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 1009	. . . . . . . . . . 11 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> t e. Q. )
15	14	ad2antrr 480	. . . . . . . . . 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 520	. . . . . . . . . . 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 7363	. . . . . . . . . . 11 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
18		recclnq 7333	. . . . . . . . . . 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 7322	. . . . . . . . . . 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 6011	. . . . . . . . 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 7535	. . . . . . . 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 7537	. . . . . . . 8 |- <P Or P.
27		ltrelpr 7446	. . . . . . . 8 |- <P C_ ( P. X. P. )
28	26, 27	sotri 4999	. . . . . . 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 417	. . . . . 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 2568	. . . 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 3758	. . . . . . . . . . 11 |- ( b = r -> <. b , 1o >. = <. r , 1o >. )
34	33	eceq1d 6537	. . . . . . . . . 10 |- ( b = r -> [ <. b , 1o >. ] ~Q = [ <. r , 1o >. ] ~Q )
35	34	fveq2d 5490	. . . . . . . . 9 |- ( b = r -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. r , 1o >. ] ~Q ) )
36	35	oveq2d 5858	. . . . . . . 8 |- ( b = r -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	breq2d 3994	. . . . . . 7 |- ( b = r -> ( p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
38	37	abbidv 2284	. . . . . 6 |- ( b = r -> { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
39	36	breq1d 3992	. . . . . . 7 |- ( b = r -> ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
40	39	abbidv 2284	. . . . . 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 3766	. . . . 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 5486	. . . . 5 |- ( b = r -> ( F ` b ) = ( F ` r ) )
43	41, 42	breq12d 3995	. . . 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 2693	. . 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 7626	. 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 414	1 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   1oc1o 6377   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   Q.cnq 7221   +Q cplq 7223   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  caucvgprprlemrnd  7642