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Theorem rge0scvg 28085
Description: Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 14441 (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 11036 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10812 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
1  e.  ZZ )
3 rge0ssre 11549 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
4 fss 5647 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : NN --> RR )
53, 4mpan2 669 . . . . . 6  |-  ( F : NN --> ( 0 [,) +oo )  ->  F : NN --> RR )
65ffvelrnda 5933 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
71, 2, 6serfre 12039 . . . 4  |-  ( F : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  F ) : NN --> RR )
8 frn 5645 . . . 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 463 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
11 1nn 10463 . . . . 5  |-  1  e.  NN
12 fdm 5643 . . . . 5  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  dom  seq 1
(  +  ,  F
)  =  NN )
1311, 12syl5eleqr 2477 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq 1 (  +  ,  F ) )
14 ne0i 3717 . . . . 5  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
15 dm0rn0 5132 . . . . . 6  |-  ( dom 
seq 1 (  +  ,  F )  =  (/) 
<->  ran  seq 1 (  +  ,  F )  =  (/) )
1615necon3bii 2650 . . . . 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 463 . 2  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
20 1zzd 10812 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  1  e.  ZZ )
21 climdm 13379 . . . . . . 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 464 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
247adantr 463 . . . . . 6  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  seq 1 (  +  ,  F ) : NN --> RR )
2524ffvelrnda 5933 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  e.  RR )
261, 20, 23, 25climrecl 13408 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  (  ~~>  `  seq 1 (  +  ,  F ) )  e.  RR )
27 simpr 459 . . . . . 6  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  k  e.  NN )
2823adantr 463 . . . . . 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 5931 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  ( 0 [,) +oo ) )
313, 30sseldi 3415 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
3229, 31sylancom 665 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
33 elrege0 11548 . . . . . . . . . 10  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  <->  ( ( F `
 j )  e.  RR  /\  0  <_ 
( F `  j
) ) )
3433simprbi 462 . . . . . . . . 9  |-  ( ( F `  j )  e.  ( 0 [,) +oo )  ->  0  <_ 
( F `  j
) )
3530, 34syl 16 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3635adantlr 712 . . . . . . 7  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  j  e.  NN )  ->  0  <_  ( F `  j ) )
3736adantlr 712 . . . . . 6  |-  ( ( ( ( F : NN
--> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
381, 27, 28, 32, 37climserle 13487 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq 1
(  +  ,  F
) ) )
3938ralrimiva 2796 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
40 breq2 4371 . . . . . 6  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
4140ralbidv 2821 . . . . 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 3135 . . . 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 659 . . 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 5639 . . . . . 6  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
45 breq1 4370 . . . . . . 7  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
4645ralrn 5936 . . . . . 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 2893 . . . 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 463 . . 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 10419 . 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 1226 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733    C_ wss 3389   (/)c0 3711   class class class wbr 4367   dom cdm 4913   ran crn 4914    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406   +oocpnf 9536    < clt 9539    <_ cle 9540   NNcn 10452   [,)cico 11452    seqcseq 12010    ~~> cli 13309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-ico 11456  df-fz 11594  df-fl 11828  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314
This theorem is referenced by: (None)
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