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Theorem inffz 13072
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 503 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
21zred 9127 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  RR )
3 simprr 504 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
43zred 9127 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  RR )
52, 4lttri3d 7842 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  =  y  <->  ( -.  x  <  y  /\  -.  y  <  x ) ) )
6 eluzel2 9283 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
7 eluzfz1 9762 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
8 elfzle1 9758 . . . 4  |-  ( z  e.  ( M ... N )  ->  M  <_  z )
98adantl 273 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  z  e.  ( M ... N
) )  ->  M  <_  z )
106zred 9127 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  RR )
11 elfzelz 9757 . . . . 5  |-  ( z  e.  ( M ... N )  ->  z  e.  ZZ )
1211zred 9127 . . . 4  |-  ( z  e.  ( M ... N )  ->  z  e.  RR )
13 lenlt 7804 . . . 4  |-  ( ( M  e.  RR  /\  z  e.  RR )  ->  ( M  <_  z  <->  -.  z  <  M ) )
1410, 12, 13syl2an 285 . . 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 6880 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 1314    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740  infcinf 6836   RRcr 7583    < clt 7764    <_ cle 7765   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696  ax-pre-apti 7699
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-neg 7900  df-z 9009  df-uz 9279  df-fz 9742
This theorem is referenced by: (None)
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