Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inffz Structured version   Unicode version

Theorem inffz 29349
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
Assertion
Ref Expression
inffz  |-  ( N  e.  ( ZZ>= `  M
)  ->  sup (
( M ... N
) ,  ZZ ,  `'  <  )  =  M )

Proof of Theorem inffz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 zssre 10867 . . . . 5  |-  ZZ  C_  RR
2 ltso 9654 . . . . 5  |-  <  Or  RR
3 soss 4807 . . . . 5  |-  ( ZZ  C_  RR  ->  (  <  Or  RR  ->  <  Or  ZZ ) )
41, 2, 3mp2 9 . . . 4  |-  <  Or  ZZ
5 cnvso 5529 . . . 4  |-  (  < 
Or  ZZ  <->  `'  <  Or  ZZ )
64, 5mpbi 208 . . 3  |-  `'  <  Or  ZZ
76a1i 11 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  `'  <  Or  ZZ )
8 eluzel2 11087 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
9 eluzfz1 11696 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
10 elfzle1 11692 . . . . 5  |-  ( x  e.  ( M ... N )  ->  M  <_  x )
1110adantl 464 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  ( M ... N
) )  ->  M  <_  x )
128zred 10965 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
13 elfzelz 11691 . . . . . 6  |-  ( x  e.  ( M ... N )  ->  x  e.  ZZ )
1413zred 10965 . . . . 5  |-  ( x  e.  ( M ... N )  ->  x  e.  RR )
15 lenlt 9652 . . . . 5  |-  ( ( M  e.  RR  /\  x  e.  RR )  ->  ( M  <_  x  <->  -.  x  <  M ) )
1612, 14, 15syl2an 475 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  ( M ... N
) )  ->  ( M  <_  x  <->  -.  x  <  M ) )
1711, 16mpbid 210 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  ( M ... N
) )  ->  -.  x  <  M )
18 brcnvg 5172 . . . . 5  |-  ( ( M  e.  ZZ  /\  x  e.  ( M ... N ) )  -> 
( M `'  <  x  <-> 
x  <  M )
)
1918notbid 292 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  ( M ... N ) )  -> 
( -.  M `'  <  x  <->  -.  x  <  M ) )
208, 19sylan 469 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  ( M ... N
) )  ->  ( -.  M `'  <  x  <->  -.  x  <  M ) )
2117, 20mpbird 232 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  ( M ... N
) )  ->  -.  M `'  <  x )
227, 8, 9, 21supmax 7915 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  sup (
( M ... N
) ,  ZZ ,  `'  <  )  =  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439    Or wor 4788   `'ccnv 4987   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480    < clt 9617    <_ cle 9618   ZZcz 10860   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-pre-lttri 9555  ax-pre-lttrn 9556
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-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  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-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9799  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator