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Theorem rge0scvg 26401
Description: Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 14003 (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 10917 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1z 10697 . . . . . 6  |-  1  e.  ZZ
32a1i 11 . . . . 5  |-  ( F : NN --> ( 0 [,) +oo )  -> 
1  e.  ZZ )
4 rge0ssre 11414 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
5 fss 5588 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : NN --> RR )
64, 5mpan2 671 . . . . . 6  |-  ( F : NN --> ( 0 [,) +oo )  ->  F : NN --> RR )
76ffvelrnda 5864 . . . . 5  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
81, 3, 7serfre 11856 . . . 4  |-  ( F : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  F ) : NN --> RR )
9 frn 5586 . . . 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 10354 . . . . 5  |-  1  e.  NN
13 fdm 5584 . . . . 5  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  dom  seq 1
(  +  ,  F
)  =  NN )
1412, 13syl5eleqr 2530 . . . 4  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  1  e.  dom  seq 1 (  +  ,  F ) )
15 ne0i 3664 . . . . 5  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
16 dm0rn0 5077 . . . . . 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 13053 . . . . . . 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 5864 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  e.  RR )
271, 21, 24, 26climrecl 13082 . . . 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 5862 . . . . . . . 8  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  j  e.  NN )  ->  ( F `  j
)  e.  ( 0 [,) +oo ) )
334, 32sseldi 3375 . . . . . . 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 11413 . . . . . . . . . 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 13161 . . . . 5  |-  ( ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 k )  <_ 
(  ~~>  `  seq 1
(  +  ,  F
) ) )
4140ralrimiva 2820 . . . 4  |-  ( ( F : NN --> ( 0 [,) +oo )  /\  seq 1 (  +  ,  F )  e.  dom  ~~>  )  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
42 breq2 4317 . . . . . 6  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
4342ralbidv 2756 . . . . 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 3094 . . . 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 5580 . . . . . 6  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
47 breq1 4316 . . . . . . 7  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
4847ralrn 5867 . . . . . 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 2757 . . . 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 10311 . 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 2620   A.wral 2736   E.wrex 2737    C_ wss 3349   (/)c0 3658   class class class wbr 4313   dom cdm 4861   ran crn 4862    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   supcsup 7711   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306   +oocpnf 9436    < clt 9439    <_ cle 9440   NNcn 10343   ZZcz 10667   [,)cico 11323    seqcseq 11827    ~~> cli 12983
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ico 11327  df-fz 11459  df-fl 11663  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-rlim 12988
This theorem is referenced by: (None)
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