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Theorem fseqsupcl 11236
Description: The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fseqsupcl  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )

Proof of Theorem fseqsupcl
StepHypRef Expression
1 frn 5530 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 453 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fzfi 11231 . . . 4  |-  ( M ... N )  e. 
Fin
4 ffn 5524 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
54adantl 453 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  F  Fn  ( M ... N ) )
6 dffn4 5592 . . . . 5  |-  ( F  Fn  ( M ... N )  <->  F :
( M ... N
) -onto-> ran  F )
75, 6sylib 189 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  F : ( M ... N )
-onto->
ran  F )
8 fofi 7321 . . . 4  |-  ( ( ( M ... N
)  e.  Fin  /\  F : ( M ... N ) -onto-> ran  F
)  ->  ran  F  e. 
Fin )
93, 7, 8sylancr 645 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  e.  Fin )
10 fdm 5528 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
1110adantl 453 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =  ( M ... N ) )
12 simpl 444 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  N  e.  (
ZZ>= `  M ) )
13 fzn0 10995 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
1412, 13sylibr 204 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ( M ... N )  =/=  (/) )
1511, 14eqnetrd 2561 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =/=  (/) )
16 dm0rn0 5019 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1716necon3bii 2575 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1815, 17sylib 189 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
19 ltso 9082 . . . 4  |-  <  Or  RR
20 fisupcl 7398 . . . 4  |-  ( (  <  Or  RR  /\  ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
2119, 20mpan 652 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR )  ->  sup ( ran  F ,  RR ,  <  )  e. 
ran  F )
229, 18, 2, 21syl3anc 1184 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
232, 22sseldd 3285 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543    C_ wss 3256   (/)c0 3564    Or wor 4436   dom cdm 4811   ran crn 4812    Fn wfn 5382   -->wf 5383   -onto->wfo 5385   ` cfv 5387  (class class class)co 6013   Fincfn 7038   supcsup 7373   RRcr 8915    < clt 9046   ZZ>=cuz 10413   ...cfz 10968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969
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