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Theorem fseqsupcl 11804
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 5570 . . 3  |-  ( F : ( M ... N ) --> RR  ->  ran 
F  C_  RR )
21adantl 466 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  C_  RR )
3 fzfi 11799 . . . 4  |-  ( M ... N )  e. 
Fin
4 ffn 5564 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  F  Fn  ( M ... N ) )
54adantl 466 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  F  Fn  ( M ... N ) )
6 dffn4 5631 . . . . 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 7602 . . . 4  |-  ( ( ( M ... N
)  e.  Fin  /\  F : ( M ... N ) -onto-> ran  F
)  ->  ran  F  e. 
Fin )
93, 7, 8sylancr 663 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  e.  Fin )
10 fdm 5568 . . . . . 6  |-  ( F : ( M ... N ) --> RR  ->  dom 
F  =  ( M ... N ) )
1110adantl 466 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =  ( M ... N ) )
12 simpl 457 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  N  e.  (
ZZ>= `  M ) )
13 fzn0 11469 . . . . . 6  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
1412, 13sylibr 212 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ( M ... N )  =/=  (/) )
1511, 14eqnetrd 2631 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  dom  F  =/=  (/) )
16 dm0rn0 5061 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1716necon3bii 2645 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1815, 17sylib 196 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  ran  F  =/=  (/) )
19 ltso 9460 . . . 4  |-  <  Or  RR
20 fisupcl 7722 . . . 4  |-  ( (  <  Or  RR  /\  ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR ) )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
2119, 20mpan 670 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  F  =/=  (/)  /\  ran  F 
C_  RR )  ->  sup ( ran  F ,  RR ,  <  )  e. 
ran  F )
229, 18, 2, 21syl3anc 1218 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
232, 22sseldd 3362 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611    C_ wss 3333   (/)c0 3642    Or wor 4645   dom cdm 4845   ran crn 4846    Fn wfn 5418   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   Fincfn 7315   supcsup 7695   RRcr 9286    < clt 9423   ZZ>=cuz 10866   ...cfz 11442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443
This theorem is referenced by: (None)
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