Theorem caucvgprlemcl 6647

Description: Lemma for caucvgpr 6653. 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 5121	. . . . . . 7 |- ( j = a -> ( F ` j) = ( F ` a))
5	4	breq2d 3767	. . . . . 6 |- ( j = a -> (A <Q ( F ` j) <-> A <Q ( F ` a)))
6	5	cbvralv 2527	. . . . 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 3540	. . . . . . . . . . . . 13 |- ( j = a -> <. j , 1o >. = <. a , 1o >.)
10	9	eceq1d 6078	. . . . . . . . . . . 12 |- ( j = a -> [ <. j , 1o >.] ~Q = [ <. a , 1o >.] ~Q )
11	10	fveq2d 5125	. . . . . . . . . . 11 |- ( j = a -> ( *Q ` [ <. j , 1o >.] ~Q ) = ( *Q ` [ <. a , 1o >.] ~Q ))
12	11	oveq2d 5471	. . . . . . . . . 10 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >.] ~Q )) = ( l +Q ( *Q ` [ <. a , 1o >.] ~Q )))
13	12, 4	breq12d 3768	. . . . . . . . 9 |- ( j = a -> (( l +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q ( F ` j) <-> ( l +Q ( *Q ` [ <. a , 1o >.] ~Q )) <Q ( F ` a)))
14	13	cbvrexv 2528	. . . . . . . 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 2541	. . . . . 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 5473	. . . . . . . . . 10 |- ( j = a -> (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) = (( F ` a) +Q ( *Q ` [ <. a , 1o >.] ~Q )))
18	17	breq1d 3765	. . . . . . . . 9 |- ( j = a -> ((( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q u <-> (( F ` a) +Q ( *Q ` [ <. a , 1o >.] ~Q )) <Q u))
19	18	cbvrexv 2528	. . . . . . . 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 2541	. . . . . 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 3545	. . . . 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 2057	. . . 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 6639	. . 3 |- (ph -> (E. s e. Q. s e. ( 1st ` L) /\ E. r e. Q. r e. ( 2nd ` L)))
25		ssrab2 3019	. . . . . 6 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q ( F ` j) } C_ Q.
26		nqex 6347	. . . . . . 7 |- Q. e. _V
27	26	elpw2 3902	. . . . . 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 3019	. . . . . 6 |- {u e. Q. | E. j e. N. (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q u } C_ Q.
30	26	elpw2 3902	. . . . . 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 4319	. . . . 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 2107	. . 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 6644	. . 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 3758	. . . . . . 7 |- ( n = c -> ( n <N k <-> c <N k))
38		fveq2 5121	. . . . . . . . 9 |- ( n = c -> ( F ` n) = ( F ` c))
39		opeq1 3540	. . . . . . . . . . . 12 |- ( n = c -> <. n , 1o >. = <. c , 1o >.)
40	39	eceq1d 6078	. . . . . . . . . . 11 |- ( n = c -> [ <. n , 1o >.] ~Q = [ <. c , 1o >.] ~Q )
41	40	fveq2d 5125	. . . . . . . . . 10 |- ( n = c -> ( *Q ` [ <. n , 1o >.] ~Q ) = ( *Q ` [ <. c , 1o >.] ~Q ))
42	41	oveq2d 5471	. . . . . . . . 9 |- ( n = c -> (( F ` k) +Q ( *Q ` [ <. n , 1o >.] ~Q )) = (( F ` k) +Q ( *Q ` [ <. c , 1o >.] ~Q )))
43	38, 42	breq12d 3768	. . . . . . . 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 5473	. . . . . . . . 9 |- ( n = c -> (( F ` n) +Q ( *Q ` [ <. n , 1o >.] ~Q )) = (( F ` c) +Q ( *Q ` [ <. c , 1o >.] ~Q )))
45	44	breq2d 3767	. . . . . . . 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 3759	. . . . . . 7 |- ( k = d -> ( c <N k <-> c <N d))
49		fveq2 5121	. . . . . . . . . 10 |- ( k = d -> ( F ` k) = ( F ` d))
50	49	oveq1d 5470	. . . . . . . . 9 |- ( k = d -> (( F ` k) +Q ( *Q ` [ <. c , 1o >.] ~Q )) = (( F ` d) +Q ( *Q ` [ <. c , 1o >.] ~Q )))
51	50	breq2d 3767	. . . . . . . 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 3765	. . . . . . . 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 2535	. . . . 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 6645	. . 3 |- (ph -> A. s e. Q. -. ( s e. ( 1st ` L) /\ s e. ( 2nd ` L)))
58	1, 2, 7, 23	caucvgprlemloc 6646	. . 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 1083	. 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 6459	. 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 628   /\ w3a 884   = wceq 1242   e. wcel 1390  A.wral 2300  E.wrex 2301   {crab 2304   C_ wss 2911   ~Pcpw 3351   <.cop 3370   class class class wbr 3755   X. cxp 4286   -->wf 4841   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   1oc1o 5933  [cec 6040   N.cnpi 6256   <N clti 6259   ~Q ceq 6263   Q.cnq 6264   +Q cplq 6266   *Qcrq 6268   <Q cltq 6269   P.cnp 6275

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449

This theorem is referenced by:  caucvgprlemladdfu  6648  caucvgprlemladdrl  6649  caucvgprlem1  6650  caucvgprlem2  6651  caucvgpr  6653