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Theorem fseqsupubi 12091
Description: The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)
Assertion
Ref Expression
fseqsupubi  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  <_  sup ( ran  F ,  RR ,  <  ) )

Proof of Theorem fseqsupubi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frn 5743 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 466 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fdm 5741 . . 3  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
4 ne0i 3799 . . . 4  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =/=  (/) )
5 dm0rn0 5229 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
6 eqeq1 2461 . . . . . . 7  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/)  <->  ( M ... N )  =  (/) ) )
76biimpd 207 . . . . . 6  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
85, 7syl5bir 218 . . . . 5  |-  ( dom 
F  =  ( M ... N )  -> 
( ran  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
98necon3d 2681 . . . 4  |-  ( dom 
F  =  ( M ... N )  -> 
( ( M ... N )  =/=  (/)  ->  ran  F  =/=  (/) ) )
104, 9mpan9 469 . . 3  |-  ( ( K  e.  ( M ... N )  /\  dom  F  =  ( M ... N ) )  ->  ran  F  =/=  (/) )
113, 10sylan2 474 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
12 fsequb2 12089 . . 3  |-  ( F : ( M ... N ) --> RR  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
1312adantl 466 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
14 ffn 5737 . . 3  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
15 fnfvelrn 6029 . . . 4  |-  ( ( F  Fn  ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1615ancoms 453 . . 3  |-  ( ( K  e.  ( M ... N )  /\  F  Fn  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1714, 16sylan2 474 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  e.  ran  F )
18 suprub 10524 . 2  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
ran  F  y  <_  x )  /\  ( F `
 K )  e. 
ran  F )  -> 
( F `  K
)  <_  sup ( ran  F ,  RR ,  <  ) )
192, 11, 13, 17, 18syl31anc 1231 1  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  <_  sup ( ran  F ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   class class class wbr 4456   dom cdm 5008   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508    < clt 9645    <_ cle 9646   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698
This theorem is referenced by: (None)
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