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Theorem rge0scvg 27797
Description: Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 14311 (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 11120 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10896 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
1  e.  ZZ )
3 rge0ssre 11632 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
4 fss 5725 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : NN --> RR )
53, 4mpan2 671 . . . . . 6  |-  ( F : NN --> ( 0 [,) +oo )  ->  F : NN --> RR )
65ffvelrnda 6012 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
71, 2, 6serfre 12110 . . . 4  |-  ( F : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  F ) : NN --> RR )
8 frn 5723 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
97, 8syl 16 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  C_  RR )
109adantr 465 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
11 1nn 10548 . . . . 5  |-  1  e.  NN
12 fdm 5721 . . . . 5  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  dom  seq 1
(  +  ,  F
)  =  NN )
1311, 12syl5eleqr 2536 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq 1 (  +  ,  F ) )
14 ne0i 3773 . . . . 5  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
15 dm0rn0 5205 . . . . . 6  |-  ( dom 
seq 1 (  +  ,  F )  =  (/) 
<->  ran  seq 1 (  +  ,  F )  =  (/) )
1615necon3bii 2709 . . . . 5  |-  ( dom 
seq 1 (  +  ,  F )  =/=  (/) 
<->  ran  seq 1 (  +  ,  F )  =/=  (/) )
1714, 16sylib 196 . . . 4  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
187, 13, 173syl 20 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  =/=  (/) )
1918adantr 465 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
20 1zzd 10896 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  1  e.  ZZ )
21 climdm 13351 . . . . . . 7  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  <->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2221biimpi 194 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2322adantl 466 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
247adantr 465 . . . . . 6  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F ) : NN --> RR )
2524ffvelrnda 6012 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  e.  RR )
261, 20, 23, 25climrecl 13380 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  (  ~~>  `  seq 1 (  +  ,  F ) )  e.  RR )
27 simpr 461 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  k  e.  NN )
2823adantr 465 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
29 simplll 757 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  F : NN --> ( 0 [,) +oo ) )
30 ffvelrn 6010 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  ( 0 [,) +oo ) )
313, 30sseldi 3484 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
3229, 31sylancom 667 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
33 elrege0 11631 . . . . . . . . . 10  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  <->  ( ( F `
 j )  e.  RR  /\  0  <_ 
( F `  j
) ) )
3433simprbi 464 . . . . . . . . 9  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  ->  0  <_ 
( F `  j
) )
3530, 34syl 16 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3635adantlr 714 . . . . . . 7  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3736adantlr 714 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
381, 27, 28, 32, 37climserle 13459 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq 1
(  +  ,  F
) ) )
3938ralrimiva 2855 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
40 breq2 4437 . . . . . 6  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
4140ralbidv 2880 . . . . 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 ) ) ) )
4241rspcev 3194 . . . 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 )
4326, 39, 42syl2anc 661 . . 3  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k
)  <_  x )
44 ffn 5717 . . . . . 6  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
45 breq1 4436 . . . . . . 7  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
4645ralrn 6015 . . . . . 6  |-  (  seq 1 (  +  ,  F )  Fn  NN  ->  ( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
477, 44, 463syl 20 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
4847rexbidv 2952 . . . 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 ) )
4948adantr 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 ) )
5043, 49mpbird 232 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  E. x  e.  RR  A. z  e.  ran  seq 1 (  +  ,  F ) z  <_  x )
51 suprcl 10504 . 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 )
5210, 19, 50, 51syl3anc 1227 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 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792    C_ wss 3458   (/)c0 3767   class class class wbr 4433   dom cdm 4985   ran crn 4986    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   supcsup 7898   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493   +oocpnf 9623    < clt 9626    <_ cle 9627   NNcn 10537   [,)cico 11535    seqcseq 12081    ~~> cli 13281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-fz 11677  df-fl 11903  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286
This theorem is referenced by: (None)
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