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Theorem fseqsupubi 12141
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 5695 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 467 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fdm 5693 . . 3  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
4 ne0i 3710 . . . 4  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =/=  (/) )
5 dm0rn0 5013 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
6 eqeq1 2432 . . . . . . 7  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/)  <->  ( M ... N )  =  (/) ) )
76biimpd 210 . . . . . 6  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
85, 7syl5bir 221 . . . . 5  |-  ( dom 
F  =  ( M ... N )  -> 
( ran  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
98necon3d 2622 . . . 4  |-  ( dom 
F  =  ( M ... N )  -> 
( ( M ... N )  =/=  (/)  ->  ran  F  =/=  (/) ) )
104, 9mpan9 471 . . 3  |-  ( ( K  e.  ( M ... N )  /\  dom  F  =  ( M ... N ) )  ->  ran  F  =/=  (/) )
113, 10sylan2 476 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
12 fsequb2 12139 . . 3  |-  ( F : ( M ... N ) --> RR  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
1312adantl 467 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
14 ffn 5689 . . 3  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
15 fnfvelrn 5978 . . . 4  |-  ( ( F  Fn  ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1615ancoms 454 . . 3  |-  ( ( K  e.  ( M ... N )  /\  F  Fn  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1714, 16sylan2 476 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  e.  ran  F )
18 suprub 10521 . 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 1267 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 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715    C_ wss 3379   (/)c0 3704   class class class wbr 4366   dom cdm 4796   ran crn 4797    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   supcsup 7907   RRcr 9489    < clt 9626    <_ cle 9627   ...cfz 11735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736
This theorem is referenced by: (None)
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