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Theorem rge0scvg 28707
Description: Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 14808. (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 11145 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10919 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
1  e.  ZZ )
3 rge0ssre 11691 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
4 fss 5697 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : NN --> RR )
53, 4mpan2 675 . . . . . 6  |-  ( F : NN --> ( 0 [,) +oo )  ->  F : NN --> RR )
65ffvelrnda 5981 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
71, 2, 6serfre 12192 . . . 4  |-  ( F : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  F ) : NN --> RR )
8 frn 5695 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
97, 8syl 17 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  C_  RR )
109adantr 466 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
11 1nn 10571 . . . . 5  |-  1  e.  NN
12 fdm 5693 . . . . 5  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  dom  seq 1
(  +  ,  F
)  =  NN )
1311, 12syl5eleqr 2513 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq 1 (  +  ,  F ) )
14 ne0i 3710 . . . . 5  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
15 dm0rn0 5013 . . . . . 6  |-  ( dom 
seq 1 (  +  ,  F )  =  (/) 
<->  ran  seq 1 (  +  ,  F )  =  (/) )
1615necon3bii 2653 . . . . 5  |-  ( dom 
seq 1 (  +  ,  F )  =/=  (/) 
<->  ran  seq 1 (  +  ,  F )  =/=  (/) )
1714, 16sylib 199 . . . 4  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
187, 13, 173syl 18 . . 3  |-  ( F : NN --> ( 0 [,) +oo )  ->  ran  seq 1 (  +  ,  F )  =/=  (/) )
1918adantr 466 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
20 1zzd 10919 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  1  e.  ZZ )
21 climdm 13561 . . . . . . 7  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  <->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2221biimpi 197 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2322adantl 467 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
247adantr 466 . . . . . 6  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F ) : NN --> RR )
2524ffvelrnda 5981 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  e.  RR )
261, 20, 23, 25climrecl 13590 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  (  ~~>  `  seq 1 (  +  ,  F ) )  e.  RR )
27 simpr 462 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  k  e.  NN )
2823adantr 466 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
29 simplll 766 . . . . . . 7  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  F : NN --> ( 0 [,) +oo ) )
30 ffvelrn 5979 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  ( 0 [,) +oo ) )
313, 30sseldi 3405 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
3229, 31sylancom 671 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
33 elrege0 11689 . . . . . . . . . 10  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  <->  ( ( F `
 j )  e.  RR  /\  0  <_ 
( F `  j
) ) )
3433simprbi 465 . . . . . . . . 9  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  ->  0  <_ 
( F `  j
) )
3530, 34syl 17 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3635adantlr 719 . . . . . . 7  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3736adantlr 719 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
381, 27, 28, 32, 37climserle 13669 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq 1
(  +  ,  F
) ) )
3938ralrimiva 2779 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
40 breq2 4370 . . . . . 6  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
4140ralbidv 2804 . . . . 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 3125 . . . 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 665 . . 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 5689 . . . . . 6  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
45 breq1 4369 . . . . . . 7  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
4645ralrn 5984 . . . . . 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 18 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
4847rexbidv 2878 . . . 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 466 . . 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 235 . 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 10520 . 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 1264 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715    C_ wss 3379   (/)c0 3704   class class class wbr 4366   dom cdm 4796   ran crn 4797    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   supcsup 7907   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493   +oocpnf 9623    < clt 9626    <_ cle 9627   NNcn 10560   [,)cico 11588    seqcseq 12163    ~~> cli 13491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  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 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-ico 11592  df-fz 11736  df-fl 11978  df-seq 12164  df-exp 12223  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-rlim 13496
This theorem is referenced by: (None)
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