Theorem caucvgprprlemlol 7699

Description: Lemma for caucvgprpr 7713. 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 7366	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4680	. . . 4 |- ( s <Q t -> ( s e. Q. /\ t e. Q. ) )
3	2	simpld 112	. . 3 |- ( s <Q t -> s e. Q. )
4	3	3ad2ant2 1019	. 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 7686	. . . . . 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 275	. . . . 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 1020	. . . 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 1037	. . . . . . . . 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 7401	. . . . . . . . . . 11 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
11	10	adantl 277	. . . . . . . . . 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 488	. . . . . . . . . 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 114	. . . . . . . . . . . 12 |- ( s <Q t -> t e. Q. )
14	13	3ad2ant2 1019	. . . . . . . . . . 11 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> t e. Q. )
15	14	ad2antrr 488	. . . . . . . . . 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 528	. . . . . . . . . . 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 7423	. . . . . . . . . . 11 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
18		recclnq 7393	. . . . . . . . . . 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 7382	. . . . . . . . . . 11 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
21	20	adantl 277	. . . . . . . . . 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 6046	. . . . . . . . 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 147	. . . . . . . 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 7595	. . . . . . . 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 7597	. . . . . . . 8 |- <P Or P.
27		ltrelpr 7506	. . . . . . . 8 |- <P C_ ( P. X. P. )
28	26, 27	sotri 5026	. . . . . . 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 420	. . . . . 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 115	. . . . 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 2579	. . . 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 3780	. . . . . . . . . . 11 |- ( b = r -> <. b , 1o >. = <. r , 1o >. )
34	33	eceq1d 6573	. . . . . . . . . 10 |- ( b = r -> [ <. b , 1o >. ] ~Q = [ <. r , 1o >. ] ~Q )
35	34	fveq2d 5521	. . . . . . . . 9 |- ( b = r -> ( *Q ` [ <. b , 1o >. ] ~Q ) = ( *Q ` [ <. r , 1o >. ] ~Q ) )
36	35	oveq2d 5893	. . . . . . . 8 |- ( b = r -> ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	breq2d 4017	. . . . . . 7 |- ( b = r -> ( p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
38	37	abbidv 2295	. . . . . 6 |- ( b = r -> { p | p <Q ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
39	36	breq1d 4015	. . . . . . 7 |- ( b = r -> ( ( s +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
40	39	abbidv 2295	. . . . . 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 3788	. . . . 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 5517	. . . . 5 |- ( b = r -> ( F ` b ) = ( F ` r ) )
43	41, 42	breq12d 4018	. . . 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 2706	. . 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 122	. 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 7686	. 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 417	1 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   Q.cnq 7281   +Q cplq 7283   *Qcrq 7285   <Q cltq 7286   P.cnp 7292   +P. cpp 7294   <P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  caucvgprprlemrnd  7702