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Theorem fseqsupubi 11056
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 5411 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 452 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fdm 5409 . . 3  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
4 ne0i 3474 . . . 4  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =/=  (/) )
5 dm0rn0 4911 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
6 eqeq1 2302 . . . . . . 7  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/)  <->  ( M ... N )  =  (/) ) )
76biimpd 198 . . . . . 6  |-  ( dom 
F  =  ( M ... N )  -> 
( dom  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
85, 7syl5bir 209 . . . . 5  |-  ( dom 
F  =  ( M ... N )  -> 
( ran  F  =  (/) 
->  ( M ... N
)  =  (/) ) )
98necon3d 2497 . . . 4  |-  ( dom 
F  =  ( M ... N )  -> 
( ( M ... N )  =/=  (/)  ->  ran  F  =/=  (/) ) )
104, 9mpan9 455 . . 3  |-  ( ( K  e.  ( M ... N )  /\  dom  F  =  ( M ... N ) )  ->  ran  F  =/=  (/) )
113, 10sylan2 460 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
12 fsequb2 11054 . . 3  |-  ( F : ( M ... N ) --> RR  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
1312adantl 452 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  E. x  e.  RR  A. y  e.  ran  F  y  <_  x )
14 ffn 5405 . . 3  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
15 fnfvelrn 5678 . . . 4  |-  ( ( F  Fn  ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1615ancoms 439 . . 3  |-  ( ( K  e.  ( M ... N )  /\  F  Fn  ( M ... N ) )  -> 
( F `  K
)  e.  ran  F
)
1714, 16sylan2 460 . 2  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  e.  ran  F )
18 suprub 9731 . 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 1185 1  |-  ( ( K  e.  ( M ... N )  /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  <_  sup ( ran  F ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752    < clt 8883    <_ cle 8884   ...cfz 10798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799
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