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Theorem inffz 14101
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 526 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
21zred 9334 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  RR )
3 simprr 527 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
43zred 9334 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  RR )
52, 4lttri3d 8034 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  =  y  <->  ( -.  x  <  y  /\  -.  y  <  x ) ) )
6 eluzel2 9492 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
7 eluzfz1 9987 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
8 elfzle1 9983 . . . 4  |-  ( z  e.  ( M ... N )  ->  M  <_  z )
98adantl 275 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  z  e.  ( M ... N
) )  ->  M  <_  z )
106zred 9334 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
11 elfzelz 9981 . . . . 5  |-  ( z  e.  ( M ... N )  ->  z  e.  ZZ )
1211zred 9334 . . . 4  |-  ( z  e.  ( M ... N )  ->  z  e.  RR )
13 lenlt 7995 . . . 4  |-  ( ( M  e.  RR  /\  z  e.  RR )  ->  ( M  <_  z  <->  -.  z  <  M ) )
1410, 12, 13syl2an 287 . . 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 7004 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 1348    e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853  infcinf 6960   RRcr 7773    < clt 7954    <_ cle 7955   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-apti 7889
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-neg 8093  df-z 9213  df-uz 9488  df-fz 9966
This theorem is referenced by: (None)
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