Theorem caucvgprlemcl 6774

Description: Lemma for caucvgpr 6780. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-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
caucvgprlemcl	|- ( ph -> L e. P. )

Distinct variable groups:   A, j   j, F, l   u, F, j   n, F, k   j, k, L   k, n

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

Proof of Theorem caucvgprlemcl

Dummy variables s a c d r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . 4 |- ( ph -> F : N. --> Q. )
2		caucvgpr.cau	. . . 4 |- ( 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 ) ) ) ) )
3		caucvgpr.bnd	. . . . 5 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		fveq2 5178	. . . . . . 7 |- ( j = a -> ( F ` j ) = ( F ` a ) )
5	4	breq2d 3776	. . . . . 6 |- ( j = a -> ( A <Q ( F ` j ) <-> A <Q ( F ` a ) ) )
6	5	cbvralv 2533	. . . . 5 |- ( A. j e. N. A <Q ( F ` j ) <-> A. a e. N. A <Q ( F ` a ) )
7	3, 6	sylib 127	. . . 4 |- ( ph -> A. a e. N. A <Q ( F ` a ) )
8		caucvgpr.lim	. . . . 5 |- 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		opeq1 3549	. . . . . . . . . . . . 13 |- ( j = a -> <. j , 1o >. = <. a , 1o >. )
10	9	eceq1d 6142	. . . . . . . . . . . 12 |- ( j = a -> [ <. j , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
11	10	fveq2d 5182	. . . . . . . . . . 11 |- ( j = a -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
12	11	oveq2d 5528	. . . . . . . . . 10 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
13	12, 4	breq12d 3777	. . . . . . . . 9 |- ( j = a -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) ) )
14	13	cbvrexv 2534	. . . . . . . 8 |- ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) )
15	14	a1i 9	. . . . . . 7 |- ( l e. Q. -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) ) )
16	15	rabbiia 2547	. . . . . 6 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } = { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) }
17	4, 11	oveq12d 5530	. . . . . . . . . 10 |- ( j = a -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
18	17	breq1d 3774	. . . . . . . . 9 |- ( j = a -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u ) )
19	18	cbvrexv 2534	. . . . . . . 8 |- ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. a e. N. ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u )
20	19	a1i 9	. . . . . . 7 |- ( u e. Q. -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. a e. N. ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u ) )
21	20	rabbiia 2547	. . . . . 6 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } = { u e. Q. | E. a e. N. ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u }
22	16, 21	opeq12i 3554	. . . . 5 |- <. { 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. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) } , { u e. Q. | E. a e. N. ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >.
23	8, 22	eqtri 2060	. . . 4 |- L = <. { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) } , { u e. Q. | E. a e. N. ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >.
24	1, 2, 7, 23	caucvgprlemm 6766	. . 3 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
25		ssrab2 3025	. . . . . 6 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } C_ Q.
26		nqex 6461	. . . . . . 7 |- Q. e. _V
27	26	elpw2 3911	. . . . . 6 |- ( { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. ~P Q. <-> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } C_ Q. )
28	25, 27	mpbir 134	. . . . 5 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. ~P Q.
29		ssrab2 3025	. . . . . 6 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } C_ Q.
30	26	elpw2 3911	. . . . . 6 |- ( { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. ~P Q. <-> { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } C_ Q. )
31	29, 30	mpbir 134	. . . . 5 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. ~P Q.
32		opelxpi 4376	. . . . 5 |- ( ( { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. ~P Q. /\ { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. ~P Q. ) -> <. { 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 } >. e. ( ~P Q. X. ~P Q. ) )
33	28, 31, 32	mp2an 402	. . . 4 |- <. { 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 } >. e. ( ~P Q. X. ~P Q. )
34	8, 33	eqeltri 2110	. . 3 |- L e. ( ~P Q. X. ~P Q. )
35	24, 34	jctil 295	. 2 |- ( ph -> ( L e. ( ~P Q. X. ~P Q. ) /\ ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) ) )
36	1, 2, 7, 23	caucvgprlemrnd 6771	. . 3 |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )
37		breq1 3767	. . . . . . 7 |- ( n = c -> ( n <N k <-> c <N k ) )
38		fveq2 5178	. . . . . . . . 9 |- ( n = c -> ( F ` n ) = ( F ` c ) )
39		opeq1 3549	. . . . . . . . . . . 12 |- ( n = c -> <. n , 1o >. = <. c , 1o >. )
40	39	eceq1d 6142	. . . . . . . . . . 11 |- ( n = c -> [ <. n , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
41	40	fveq2d 5182	. . . . . . . . . 10 |- ( n = c -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
42	41	oveq2d 5528	. . . . . . . . 9 |- ( n = c -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
43	38, 42	breq12d 3777	. . . . . . . 8 |- ( n = c -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
44	38, 41	oveq12d 5530	. . . . . . . . 9 |- ( n = c -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
45	44	breq2d 3776	. . . . . . . 8 |- ( n = c -> ( ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
46	43, 45	anbi12d 442	. . . . . . 7 |- ( n = c -> ( ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) <-> ( ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) )
47	37, 46	imbi12d 223	. . . . . 6 |- ( n = c -> ( ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) <-> ( c <N k -> ( ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) ) )
48		breq2 3768	. . . . . . 7 |- ( k = d -> ( c <N k <-> c <N d ) )
49		fveq2 5178	. . . . . . . . . 10 |- ( k = d -> ( F ` k ) = ( F ` d ) )
50	49	oveq1d 5527	. . . . . . . . 9 |- ( k = d -> ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) = ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
51	50	breq2d 3776	. . . . . . . 8 |- ( k = d -> ( ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <-> ( F ` c ) <Q ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
52	49	breq1d 3774	. . . . . . . 8 |- ( k = d -> ( ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) <-> ( F ` d ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) )
53	51, 52	anbi12d 442	. . . . . . 7 |- ( k = d -> ( ( ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) <-> ( ( F ` c ) <Q ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` d ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) )
54	48, 53	imbi12d 223	. . . . . 6 |- ( k = d -> ( ( c <N k -> ( ( F ` c ) <Q ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) <-> ( c <N d -> ( ( F ` c ) <Q ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` d ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) ) )
55	47, 54	cbvral2v 2541	. . . . 5 |- ( 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 ) ) ) ) <-> A. c e. N. A. d e. N. ( c <N d -> ( ( F ` c ) <Q ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` d ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) )
56	2, 55	sylib 127	. . . 4 |- ( ph -> A. c e. N. A. d e. N. ( c <N d -> ( ( F ` c ) <Q ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) /\ ( F ` d ) <Q ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) ) ) )
57	1, 56, 7, 23	caucvgprlemdisj 6772	. . 3 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
58	1, 2, 7, 23	caucvgprlemloc 6773	. . 3 |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
59	36, 57, 58	3jca 1084	. 2 |- ( ph -> ( ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) /\ A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) /\ A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) )
60		elnp1st2nd 6574	. 2 |- ( L e. P. <-> ( ( L e. ( ~P Q. X. ~P Q. ) /\ ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) ) /\ ( ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) /\ A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) /\ A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) ) )
61	35, 59, 60	sylanbrc 394	1 |- ( ph -> L e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   C_ wss 2917   ~Pcpw 3359   <.cop 3378   class class class wbr 3764   X. cxp 4343   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   1oc1o 5994   [cec 6104   N.cnpi 6370   <N clti 6373   ~Q ceq 6377   Q.cnq 6378   +Q cplq 6380   *Qcrq 6382   <Q cltq 6383   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  caucvgprlemladdfu  6775  caucvgprlemladdrl  6776  caucvgprlem1  6777  caucvgprlem2  6778  caucvgpr  6780