MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzinfi Structured version   Unicode version

Theorem uzinfi 11240
Description: Extract the lower bound of an upper set of integers as its infimum. (Contributed by NM, 7-Oct-2005.) (Revised by AV, 4-Sep-2020.)
Hypothesis
Ref Expression
uzinfi.1  |-  M  e.  ZZ
Assertion
Ref Expression
uzinfi  |- inf ( (
ZZ>= `  M ) ,  RR ,  <  )  =  M

Proof of Theorem uzinfi
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 uzinfi.1 . 2  |-  M  e.  ZZ
2 ltso 9716 . . . 4  |-  <  Or  RR
32a1i 11 . . 3  |-  ( M  e.  ZZ  ->  <  Or  RR )
4 zre 10943 . . 3  |-  ( M  e.  ZZ  ->  M  e.  RR )
5 uzid 11175 . . 3  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
6 eluz2 11167 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <_ 
k ) )
74adantr 467 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  M  e.  RR )
8 zre 10943 . . . . . . . . 9  |-  ( k  e.  ZZ  ->  k  e.  RR )
98adantl 468 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  k  e.  RR )
107, 9lenltd 9783 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( M  <_  k  <->  -.  k  <  M ) )
1110biimp3a 1365 . . . . . 6  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <_  k )  ->  -.  k  <  M )
1211a1d 27 . . . . 5  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <_  k )  ->  ( M  e.  ZZ  ->  -.  k  <  M ) )
136, 12sylbi 199 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( M  e.  ZZ  ->  -.  k  <  M ) )
1413impcom 432 . . 3  |-  ( ( M  e.  ZZ  /\  k  e.  ( ZZ>= `  M ) )  ->  -.  k  <  M )
153, 4, 5, 14infmin 8014 . 2  |-  ( M  e.  ZZ  -> inf ( (
ZZ>= `  M ) ,  RR ,  <  )  =  M )
161, 15ax-mp 5 1  |- inf ( (
ZZ>= `  M ) ,  RR ,  <  )  =  M
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   class class class wbr 4421    Or wor 4771   ` cfv 5599  infcinf 7959   RRcr 9540    < clt 9677    <_ cle 9678   ZZcz 10939   ZZ>=cuz 11161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-pre-lttri 9615  ax-pre-lttrn 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-neg 9865  df-z 10940  df-uz 11162
This theorem is referenced by:  nninf  11242  nn0inf  11243
  Copyright terms: Public domain W3C validator