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Theorem fseqsupcl 12072
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 5719 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 464 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fzfi 12067 . . . 4  |-  ( M ... N )  e. 
Fin
4 ffn 5713 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
54adantl 464 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  F  Fn  ( M ... N ) )
6 dffn4 5783 . . . . 5  |-  ( F  Fn  ( M ... N )  <->  F :
( M ... N
) -onto-> ran  F )
75, 6sylib 196 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  F : ( M ... N )
-onto->
ran  F )
8 fofi 7798 . . . 4  |-  ( ( ( M ... N
)  e.  Fin  /\  F : ( M ... N ) -onto-> ran  F
)  ->  ran  F  e. 
Fin )
93, 7, 8sylancr 661 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  e.  Fin )
10 fdm 5717 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
1110adantl 464 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =  ( M ... N ) )
12 simpl 455 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  N  e.  (
ZZ>= `  M ) )
13 fzn0 11703 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
1412, 13sylibr 212 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ( M ... N )  =/=  (/) )
1511, 14eqnetrd 2747 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =/=  (/) )
16 dm0rn0 5208 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1716necon3bii 2722 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1815, 17sylib 196 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
19 ltso 9654 . . . 4  |-  <  Or  RR
20 fisupcl 7919 . . . 4  |-  ( (  <  Or  RR  /\  ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
2119, 20mpan 668 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR )  ->  sup ( ran  F ,  RR ,  <  )  e. 
ran  F )
229, 18, 2, 21syl3anc 1226 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
232, 22sseldd 3490 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783    Or wor 4788   dom cdm 4988   ran crn 4989    Fn wfn 5565   -->wf 5566   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   Fincfn 7509   supcsup 7892   RRcr 9480    < clt 9617   ZZ>=cuz 11082   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by: (None)
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