Users' Mathboxes Mathbox for Jim Kingdon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  inffz Unicode version

Theorem inffz 12189
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
Assertion
Ref Expression
inffz  |-  ( N  e.  ( ZZ>= `  M
)  -> inf ( ( M ... N ) ,  ZZ ,  <  )  =  M )

Proof of Theorem inffz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 499 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
21zred 8929 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  RR )
3 simprr 500 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
43zred 8929 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  RR )
52, 4lttri3d 7660 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  =  y  <->  ( -.  x  <  y  /\  -.  y  <  x ) ) )
6 eluzel2 9085 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
7 eluzfz1 9506 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
8 elfzle1 9502 . . . 4  |-  ( z  e.  ( M ... N )  ->  M  <_  z )
98adantl 272 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  z  e.  ( M ... N
) )  ->  M  <_  z )
106zred 8929 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
11 elfzelz 9501 . . . . 5  |-  ( z  e.  ( M ... N )  ->  z  e.  ZZ )
1211zred 8929 . . . 4  |-  ( z  e.  ( M ... N )  ->  z  e.  RR )
13 lenlt 7622 . . . 4  |-  ( ( M  e.  RR  /\  z  e.  RR )  ->  ( M  <_  z  <->  -.  z  <  M ) )
1410, 12, 13syl2an 284 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  z  e.  ( M ... N
) )  ->  ( M  <_  z  <->  -.  z  <  M ) )
159, 14mpbid 146 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  z  e.  ( M ... N
) )  ->  -.  z  <  M )
165, 6, 7, 15infminti 6776 1  |-  ( N  e.  ( ZZ>= `  M
)  -> inf ( ( M ... N ) ,  ZZ ,  <  )  =  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666  infcinf 6732   RRcr 7410    < clt 7583    <_ cle 7584   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-pre-ltirr 7518  ax-pre-apti 7521
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sup 6733  df-inf 6734  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-neg 7717  df-z 8812  df-uz 9081  df-fz 9486
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator