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

Theorem inffz 15279
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 ∈ (ℤ𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)

Proof of Theorem inffz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ)
21zred 9405 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℝ)
3 simprr 531 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ)
43zred 9405 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℝ)
52, 4lttri3d 8102 . 2 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 = 𝑦 ↔ (¬ 𝑥 < 𝑦 ∧ ¬ 𝑦 < 𝑥)))
6 eluzel2 9563 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
7 eluzfz1 10061 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
8 elfzle1 10057 . . . 4 (𝑧 ∈ (𝑀...𝑁) → 𝑀𝑧)
98adantl 277 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → 𝑀𝑧)
106zred 9405 . . . 4 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℝ)
11 elfzelz 10055 . . . . 5 (𝑧 ∈ (𝑀...𝑁) → 𝑧 ∈ ℤ)
1211zred 9405 . . . 4 (𝑧 ∈ (𝑀...𝑁) → 𝑧 ∈ ℝ)
13 lenlt 8063 . . . 4 ((𝑀 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑀𝑧 ↔ ¬ 𝑧 < 𝑀))
1410, 12, 13syl2an 289 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝑀𝑧 ↔ ¬ 𝑧 < 𝑀))
159, 14mpbid 147 . 2 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → ¬ 𝑧 < 𝑀)
165, 6, 7, 15infminti 7056 1 (𝑁 ∈ (ℤ𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160   class class class wbr 4018  cfv 5235  (class class class)co 5896  infcinf 7012  cr 7840   < clt 8022  cle 8023  cz 9283  cuz 9558  ...cfz 10038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-pre-ltirr 7953  ax-pre-apti 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-neg 8161  df-z 9284  df-uz 9559  df-fz 10039
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator