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Theorem nnminle 12248
Description: The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12247. (Contributed by Jim Kingdon, 26-Sep-2024.)
Assertion
Ref Expression
nnminle ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem nnminle
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 dfss5 3313 . . . . . 6 (𝐴 ⊆ ℕ ↔ 𝐴 = (ℕ ∩ 𝐴))
21biimpi 119 . . . . 5 (𝐴 ⊆ ℕ → 𝐴 = (ℕ ∩ 𝐴))
3 nnuz 9479 . . . . . . 7 ℕ = (ℤ‘1)
43ineq1i 3305 . . . . . 6 (ℕ ∩ 𝐴) = ((ℤ‘1) ∩ 𝐴)
5 dfin5 3109 . . . . . 6 ((ℤ‘1) ∩ 𝐴) = {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴}
64, 5eqtri 2178 . . . . 5 (ℕ ∩ 𝐴) = {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴}
72, 6eqtrdi 2206 . . . 4 (𝐴 ⊆ ℕ → 𝐴 = {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴})
873ad2ant1 1003 . . 3 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → 𝐴 = {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴})
98infeq1d 6958 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf(𝐴, ℝ, < ) = inf({𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴}, ℝ, < ))
10 1zzd 9199 . . 3 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → 1 ∈ ℤ)
11 eqid 2157 . . 3 {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴} = {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴}
12 simp3 984 . . . 4 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → 𝐵𝐴)
1312, 8eleqtrd 2236 . . 3 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → 𝐵 ∈ {𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴})
14 eleq1w 2218 . . . . 5 (𝑥 = 𝑛 → (𝑥𝐴𝑛𝐴))
1514dcbid 824 . . . 4 (𝑥 = 𝑛 → (DECID 𝑥𝐴DECID 𝑛𝐴))
16 simpl2 986 . . . 4 (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) ∧ 𝑛 ∈ (1...𝐵)) → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)
17 elfznn 9962 . . . . 5 (𝑛 ∈ (1...𝐵) → 𝑛 ∈ ℕ)
1817adantl 275 . . . 4 (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) ∧ 𝑛 ∈ (1...𝐵)) → 𝑛 ∈ ℕ)
1915, 16, 18rspcdva 2821 . . 3 (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) ∧ 𝑛 ∈ (1...𝐵)) → DECID 𝑛𝐴)
2010, 11, 13, 19infssuzledc 11849 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf({𝑛 ∈ (ℤ‘1) ∣ 𝑛𝐴}, ℝ, < ) ≤ 𝐵)
219, 20eqbrtrd 3988 1 ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 820  w3a 963   = wceq 1335  wcel 2128  wral 2435  {crab 2439  cin 3101  wss 3102   class class class wbr 3967  cfv 5172  (class class class)co 5826  infcinf 6929  cr 7733  1c1 7735   < clt 7914  cle 7915  cn 8838  cuz 9444  ...cfz 9918
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-addcom 7834  ax-addass 7836  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-sup 6930  df-inf 6931  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-fz 9919  df-fzo 10051
This theorem is referenced by: (None)
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