Theorem caucvgprlemcl 7675

Description: Lemma for caucvgpr 7681. 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 5516	. . . . . . 7 |- ( j = a -> ( F ` j ) = ( F ` a ) )
5	4	breq2d 4016	. . . . . 6 |- ( j = a -> ( A <Q ( F ` j ) <-> A <Q ( F ` a ) ) )
6	5	cbvralv 2704	. . . . 5 |- ( A. j e. N. A <Q ( F ` j ) <-> A. a e. N. A <Q ( F ` a ) )
7	3, 6	sylib 122	. . . 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 3779	. . . . . . . . . . . . 13 |- ( j = a -> <. j , 1o >. = <. a , 1o >. )
10	9	eceq1d 6571	. . . . . . . . . . . 12 |- ( j = a -> [ <. j , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
11	10	fveq2d 5520	. . . . . . . . . . 11 |- ( j = a -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
12	11	oveq2d 5891	. . . . . . . . . 10 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
13	12, 4	breq12d 4017	. . . . . . . . 9 |- ( j = a -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) ) )
14	13	cbvrexv 2705	. . . . . . . 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 2723	. . . . . 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 5893	. . . . . . . . . 10 |- ( j = a -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
18	17	breq1d 4014	. . . . . . . . 9 |- ( j = a -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u ) )
19	18	cbvrexv 2705	. . . . . . . 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 2723	. . . . . 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 3784	. . . . 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 2198	. . . 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 7667	. . 3 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
25		ssrab2 3241	. . . . . 6 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } C_ Q.
26		nqex 7362	. . . . . . 7 |- Q. e. _V
27	26	elpw2 4158	. . . . . 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 146	. . . . 5 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. ~P Q.
29		ssrab2 3241	. . . . . 6 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } C_ Q.
30	26	elpw2 4158	. . . . . 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 146	. . . . 5 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. ~P Q.
32		opelxpi 4659	. . . . 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 426	. . . 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 2250	. . 3 |- L e. ( ~P Q. X. ~P Q. )
35	24, 34	jctil 312	. 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 7672	. . 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 4007	. . . . . . 7 |- ( n = c -> ( n <N k <-> c <N k ) )
38		fveq2 5516	. . . . . . . . 9 |- ( n = c -> ( F ` n ) = ( F ` c ) )
39		opeq1 3779	. . . . . . . . . . . 12 |- ( n = c -> <. n , 1o >. = <. c , 1o >. )
40	39	eceq1d 6571	. . . . . . . . . . 11 |- ( n = c -> [ <. n , 1o >. ] ~Q = [ <. c , 1o >. ] ~Q )
41	40	fveq2d 5520	. . . . . . . . . 10 |- ( n = c -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. c , 1o >. ] ~Q ) )
42	41	oveq2d 5891	. . . . . . . . 9 |- ( n = c -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
43	38, 42	breq12d 4017	. . . . . . . 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 5893	. . . . . . . . 9 |- ( n = c -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` c ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
45	44	breq2d 4016	. . . . . . . 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 473	. . . . . . 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 234	. . . . . 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 4008	. . . . . . 7 |- ( k = d -> ( c <N k <-> c <N d ) )
49		fveq2 5516	. . . . . . . . . 10 |- ( k = d -> ( F ` k ) = ( F ` d ) )
50	49	oveq1d 5890	. . . . . . . . 9 |- ( k = d -> ( ( F ` k ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) = ( ( F ` d ) +Q ( *Q ` [ <. c , 1o >. ] ~Q ) ) )
51	50	breq2d 4016	. . . . . . . 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 4014	. . . . . . . 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 473	. . . . . . 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 234	. . . . . 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 2717	. . . . 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 122	. . . 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 7673	. . 3 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
58	1, 2, 7, 23	caucvgprlemloc 7674	. . 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 1177	. 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 7475	. 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 417	1 |- ( ph -> L e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   C_ wss 3130   ~Pcpw 3576   <.cop 3596   class class class wbr 4004   X. cxp 4625   -->wf 5213   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   1oc1o 6410   [cec 6533   N.cnpi 7271   <N clti 7274   ~Q ceq 7278   Q.cnq 7279   +Q cplq 7281   *Qcrq 7283   <Q cltq 7284   P.cnp 7290

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465

This theorem is referenced by:  caucvgprlemladdfu  7676  caucvgprlemladdrl  7677  caucvgprlem1  7678  caucvgprlem2  7679  caucvgpr  7681