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Theorem rge0scvg 26327
Description: Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 13974 (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 10888 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1z 10668 . . . . . 6  |-  1  e.  ZZ
32a1i 11 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
1  e.  ZZ )
4 rge0ssre 11385 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
5 fss 5560 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : NN --> RR )
64, 5mpan2 671 . . . . . 6  |-  ( F : NN --> ( 0 [,) +oo )  ->  F : NN --> RR )
76ffvelrnda 5836 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
81, 3, 7serfre 11827 . . . 4  |-  ( F : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  F ) : NN --> RR )
9 frn 5558 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
108, 9syl 16 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  C_  RR )
1110adantr 465 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
12 1nn 10325 . . . . 5  |-  1  e.  NN
13 fdm 5556 . . . . 5  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  dom  seq 1
(  +  ,  F
)  =  NN )
1412, 13syl5eleqr 2524 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq 1 (  +  ,  F ) )
15 ne0i 3636 . . . . 5  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
16 dm0rn0 5048 . . . . . 6  |-  ( dom 
seq 1 (  +  ,  F )  =  (/) 
<->  ran  seq 1 (  +  ,  F )  =  (/) )
1716necon3bii 2634 . . . . 5  |-  ( dom 
seq 1 (  +  ,  F )  =/=  (/) 
<->  ran  seq 1 (  +  ,  F )  =/=  (/) )
1815, 17sylib 196 . . . 4  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
198, 14, 183syl 20 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  =/=  (/) )
2019adantr 465 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
212a1i 11 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  1  e.  ZZ )
22 climdm 13024 . . . . . . 7  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  <->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2322biimpi 194 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2423adantl 466 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
258adantr 465 . . . . . 6  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F ) : NN --> RR )
2625ffvelrnda 5836 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  e.  RR )
271, 21, 24, 26climrecl 13053 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  (  ~~>  `  seq 1 (  +  ,  F ) )  e.  RR )
28 simpr 461 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  k  e.  NN )
2924adantr 465 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
30 simplll 757 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  F : NN --> ( 0 [,) +oo ) )
31 simpr 461 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  j  e.  NN )
32 ffvelrn 5834 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  ( 0 [,) +oo ) )
334, 32sseldi 3347 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
3430, 31, 33syl2anc 661 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
35 elrege0 11384 . . . . . . . . . 10  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  <->  ( ( F `
 j )  e.  RR  /\  0  <_ 
( F `  j
) ) )
3635simprbi 464 . . . . . . . . 9  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  ->  0  <_ 
( F `  j
) )
3732, 36syl 16 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3837adantlr 714 . . . . . . 7  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3938adantlr 714 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
401, 28, 29, 34, 39climserle 13132 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq 1
(  +  ,  F
) ) )
4140ralrimiva 2793 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
42 breq2 4289 . . . . . 6  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
4342ralbidv 2729 . . . . 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 ) ) ) )
4443rspcev 3066 . . . 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 )
4527, 41, 44syl2anc 661 . . 3  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k
)  <_  x )
46 ffn 5552 . . . . . 6  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
47 breq1 4288 . . . . . . 7  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
4847ralrn 5839 . . . . . 6  |-  (  seq 1 (  +  ,  F )  Fn  NN  ->  ( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
498, 46, 483syl 20 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
5049rexbidv 2730 . . . 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 ) )
5150adantr 465 . . 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 ) )
5245, 51mpbird 232 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. z  e.  ran  seq 1 (  +  ,  F ) z  <_  x )
53 suprcl 10282 . 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 )
5411, 20, 52, 53syl3anc 1218 1  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2600   A.wral 2709   E.wrex 2710    C_ wss 3321   (/)c0 3630   class class class wbr 4285   dom cdm 4832   ran crn 4833    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   supcsup 7682   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277   +oocpnf 9407    < clt 9410    <_ cle 9411   NNcn 10314   ZZcz 10638   [,)cico 11294    seqcseq 11798    ~~> cli 12954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-ico 11298  df-fz 11430  df-fl 11634  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959
This theorem is referenced by: (None)
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