Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rge0scvg Structured version   Unicode version

Theorem rge0scvg 24337
Description: Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 13291 (Contributed by Thierry Arnoux, 28-Jul-2017.)
Assertion
Ref Expression
rge0scvg  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )

Proof of Theorem rge0scvg
Dummy variables  j 
k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 10523 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1z 10313 . . . . . 6  |-  1  e.  ZZ
32a1i 11 . . . . 5  |-  ( F : NN --> ( 0 [,)  +oo )  ->  1  e.  ZZ )
4 mnfxr 10716 . . . . . . . . 9  |-  -oo  e.  RR*
5 pnfxr 10715 . . . . . . . . 9  |-  +oo  e.  RR*
6 0re 9093 . . . . . . . . . 10  |-  0  e.  RR
7 mnflt 10724 . . . . . . . . . 10  |-  ( 0  e.  RR  ->  -oo  <  0 )
86, 7ax-mp 8 . . . . . . . . 9  |-  -oo  <  0
9 pnfge 10729 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
105, 9ax-mp 8 . . . . . . . . 9  |-  +oo  <_  +oo
11 icossioo 24135 . . . . . . . . 9  |-  ( ( (  -oo  e.  RR*  /\ 
+oo  e.  RR* )  /\  (  -oo  <  0  /\  +oo 
<_  +oo ) )  -> 
( 0 [,)  +oo )  C_  (  -oo (,)  +oo ) )
124, 5, 8, 10, 11mp4an 656 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  (  -oo (,)  +oo )
13 ioomax 10987 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
1412, 13sseqtri 3382 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR
15 fss 5601 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : NN
--> RR )
1614, 15mpan2 654 . . . . . 6  |-  ( F : NN --> ( 0 [,)  +oo )  ->  F : NN --> RR )
1716ffvelrnda 5872 . . . . 5  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
181, 3, 17serfre 11354 . . . 4  |-  ( F : NN --> ( 0 [,)  +oo )  ->  seq  1 (  +  ,  F ) : NN --> RR )
19 frn 5599 . . . 4  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  ran  seq  1
(  +  ,  F
)  C_  RR )
2018, 19syl 16 . . 3  |-  ( F : NN --> ( 0 [,)  +oo )  ->  ran  seq  1 (  +  ,  F )  C_  RR )
2120adantr 453 . 2  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq  1
(  +  ,  F
)  C_  RR )
22 1nn 10013 . . . . 5  |-  1  e.  NN
23 fdm 5597 . . . . 5  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  dom  seq  1
(  +  ,  F
)  =  NN )
2422, 23syl5eleqr 2525 . . . 4  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq  1 (  +  ,  F ) )
25 ne0i 3636 . . . . 5  |-  ( 1  e.  dom  seq  1
(  +  ,  F
)  ->  dom  seq  1
(  +  ,  F
)  =/=  (/) )
26 dm0rn0 5088 . . . . . 6  |-  ( dom 
seq  1 (  +  ,  F )  =  (/) 
<->  ran  seq  1 (  +  ,  F )  =  (/) )
2726necon3bii 2635 . . . . 5  |-  ( dom 
seq  1 (  +  ,  F )  =/=  (/) 
<->  ran  seq  1 (  +  ,  F )  =/=  (/) )
2825, 27sylib 190 . . . 4  |-  ( 1  e.  dom  seq  1
(  +  ,  F
)  ->  ran  seq  1
(  +  ,  F
)  =/=  (/) )
2918, 24, 283syl 19 . . 3  |-  ( F : NN --> ( 0 [,)  +oo )  ->  ran  seq  1 (  +  ,  F )  =/=  (/) )
3029adantr 453 . 2  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq  1
(  +  ,  F
)  =/=  (/) )
312a1i 11 . . . . 5  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  1  e.  ZZ )
32 climdm 12350 . . . . . . 7  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1 (  +  ,  F ) ) )
3332biimpi 188 . . . . . 6  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1 (  +  ,  F ) ) )
3433adantl 454 . . . . 5  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1
(  +  ,  F
) ) )
3518adantr 453 . . . . . 6  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq  1 (  +  ,  F ) : NN --> RR )
3635ffvelrnda 5872 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 k )  e.  RR )
371, 31, 34, 36climrecl 12379 . . . 4  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  (  ~~>  `  seq  1 (  +  ,  F ) )  e.  RR )
38 simpr 449 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  k  e.  NN )
3934adantr 453 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1 (  +  ,  F ) ) )
40 simplll 736 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  F : NN --> ( 0 [,)  +oo ) )
41 simpr 449 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  j  e.  NN )
42 ffvelrn 5870 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  j  e.  NN )  ->  ( F `  j )  e.  ( 0 [,)  +oo ) )
4314, 42sseldi 3348 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
4440, 41, 43syl2anc 644 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
45 elrege0 11009 . . . . . . . . . 10  |-  ( ( F `  j )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  j )  e.  RR  /\  0  <_ 
( F `  j
) ) )
4645simprbi 452 . . . . . . . . 9  |-  ( ( F `  j )  e.  ( 0 [,) 
+oo )  ->  0  <_  ( F `  j
) )
4742, 46syl 16 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  j  e.  NN )  ->  0  <_  ( F `  j
) )
4847adantlr 697 . . . . . . 7  |-  ( ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
4948adantlr 697 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
501, 38, 39, 44, 49climserle 12458 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq  1
(  +  ,  F
) ) )
5150ralrimiva 2791 . . . 4  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) )
52 breq2 4218 . . . . . 6  |-  ( x  =  (  ~~>  `  seq  1 (  +  ,  F ) )  -> 
( (  seq  1
(  +  ,  F
) `  k )  <_  x  <->  (  seq  1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq  1
(  +  ,  F
) ) ) )
5352ralbidv 2727 . . . . 5  |-  ( x  =  (  ~~>  `  seq  1 (  +  ,  F ) )  -> 
( A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x  <->  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) ) )
5453rspcev 3054 . . . 4  |-  ( ( (  ~~>  `  seq  1
(  +  ,  F
) )  e.  RR  /\ 
A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) )  ->  E. x  e.  RR  A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x )
5537, 51, 54syl2anc 644 . . 3  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k
)  <_  x )
56 ffn 5593 . . . . . 6  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  seq  1 (  +  ,  F )  Fn  NN )
57 breq1 4217 . . . . . . 7  |-  ( z  =  (  seq  1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq  1 (  +  ,  F ) `  k )  <_  x
) )
5857ralrn 5875 . . . . . 6  |-  (  seq  1 (  +  ,  F )  Fn  NN  ->  ( A. z  e. 
ran  seq  1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x ) )
5918, 56, 583syl 19 . . . . 5  |-  ( F : NN --> ( 0 [,)  +oo )  ->  ( A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x 
<-> 
A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  x
) )
6059rexbidv 2728 . . . 4  |-  ( F : NN --> ( 0 [,)  +oo )  ->  ( E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x 
<->  E. x  e.  RR  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k
)  <_  x )
)
6160adantr 453 . . 3  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  ( E. x  e.  RR  A. z  e. 
ran  seq  1 (  +  ,  F ) z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x ) )
6255, 61mpbird 225 . 2  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x )
63 suprcl 9970 . 2  |-  ( ( ran  seq  1 (  +  ,  F ) 
C_  RR  /\  ran  seq  1 (  +  ,  F )  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
6421, 30, 62, 63syl3anc 1185 1  |-  ( ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4214   dom cdm 4880   ran crn 4881    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    +oocpnf 9119    -oocmnf 9120   RR*cxr 9121    < clt 9122    <_ cle 9123   NNcn 10002   ZZcz 10284   (,)cioo 10918   [,)cico 10920    seq cseq 11325    ~~> cli 12280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ioo 10922  df-ico 10924  df-fz 11046  df-fl 11204  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285
  Copyright terms: Public domain W3C validator