Theorem caucvgprlemcl 7296

Description: Lemma for caucvgpr 7302. 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 5318	. . . . . . 7 |- ( j = a -> ( F ` j ) = ( F ` a ) )
5	4	breq2d 3863	. . . . . 6 |- ( j = a -> ( A <Q ( F ` j ) <-> A <Q ( F ` a ) ) )
6	5	cbvralv 2591	. . . . 5 |- ( A. j e. N. A <Q ( F ` j ) <-> A. a e. N. A <Q ( F ` a ) )
7	3, 6	sylib 121	. . . 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 3628	. . . . . . . . . . . . 13 |- ( j = a -> <. j , 1o >. = <. a , 1o >. )
10	9	eceq1d 6342	. . . . . . . . . . . 12 |- ( j = a -> [ <. j , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
11	10	fveq2d 5322	. . . . . . . . . . 11 |- ( j = a -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
12	11	oveq2d 5682	. . . . . . . . . 10 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
13	12, 4	breq12d 3864	. . . . . . . . 9 |- ( j = a -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) ) )
14	13	cbvrexv 2592	. . . . . . . 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 2605	. . . . . 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 5684	. . . . . . . . . 10 |- ( j = a -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
18	17	breq1d 3861	. . . . . . . . 9 |- ( j = a -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u ) )
19	18	cbvrexv 2592	. . . . . . . 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 2605	. . . . . 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 3633	. . . . 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 2109	. . . 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 7288	. . 3 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
25		ssrab2 3107	. . . . . 6 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } C_ Q.
26		nqex 6983	. . . . . . 7 |- Q. e. _V
27	26	elpw2 3999	. . . . . 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 145	. . . . 5 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. ~P Q.
29		ssrab2 3107	. . . . . 6 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } C_ Q.
30	26	elpw2 3999	. . . . . 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 145	. . . . 5 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. ~P Q.
32		opelxpi 4483	. . . . 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 418	. . . 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 2161	. . 3 |- L e. ( ~P Q. X. ~P Q. )
35	24, 34	jctil 306	. 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 7293	. . 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 3854	. . . . . . 7 |- ( n = c -> ( n <N k <-> c <N k ) )
38		fveq2 5318	. . . . . . . . 9 |- ( n = c -> ( F ` n ) = ( F ` c ) )
39		opeq1 3628	. . . . . . . . . . . 12 |- ( n = c -> <. n , 1o >. = <. c , 1o >. )
40	39	eceq1d 6342	. . . . . . . . . . 11 |- ( n = c -> [ <. n , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
41	40	fveq2d 5322	. . . . . . . . . 10 |- ( n = c -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
42	41	oveq2d 5682	. . . . . . . . 9 |- ( n = c -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
43	38, 42	breq12d 3864	. . . . . . . 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 5684	. . . . . . . . 9 |- ( n = c -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
45	44	breq2d 3863	. . . . . . . 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 458	. . . . . . 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 233	. . . . . 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 3855	. . . . . . 7 |- ( k = d -> ( c <N k <-> c <N d ) )
49		fveq2 5318	. . . . . . . . . 10 |- ( k = d -> ( F ` k ) = ( F ` d ) )
50	49	oveq1d 5681	. . . . . . . . 9 |- ( k = d -> ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) = ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
51	50	breq2d 3863	. . . . . . . 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 3861	. . . . . . . 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 458	. . . . . . 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 233	. . . . . 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 2599	. . . . 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 121	. . . 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 7294	. . 3 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
58	1, 2, 7, 23	caucvgprlemloc 7295	. . 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 1124	. 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 7096	. 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 409	1 |- ( ph -> L e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 665   /\ w3a 925   = wceq 1290   e. wcel 1439   A.wral 2360   E.wrex 2361   {crab 2364   C_ wss 3000   ~Pcpw 3433   <.cop 3453   class class class wbr 3851   X. cxp 4450   -->wf 5024   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   1oc1o 6188   [cec 6304   N.cnpi 6892   <N clti 6895   ~Q ceq 6899   Q.cnq 6900   +Q cplq 6902   *Qcrq 6904   <Q cltq 6905   P.cnp 6911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-inp 7086

This theorem is referenced by:  caucvgprlemladdfu  7297  caucvgprlemladdrl  7298  caucvgprlem1  7299  caucvgprlem2  7300  caucvgpr  7302