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Theorem zsupcl 10722
Description: Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at 𝑀 (which corresponds to the non-empty condition of classical supremum theorems), (b) decidable at each value after 𝑀, and (c) be false after 𝑗 (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
Hypotheses
Ref Expression
zsupcl.m (𝜑𝑀 ∈ ℤ)
zsupcl.sbm (𝑛 = 𝑀 → (𝜓𝜒))
zsupcl.mtru (𝜑𝜒)
zsupcl.dc ((𝜑𝑛 ∈ (ℤ𝑀)) → DECID 𝜓)
zsupcl.bnd (𝜑 → ∃𝑗 ∈ (ℤ𝑀)∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
Assertion
Ref Expression
zsupcl (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ𝑀))
Distinct variable groups:   𝜑,𝑗,𝑛   𝜓,𝑗   𝜒,𝑗,𝑛   𝑗,𝑀,𝑛
Allowed substitution hint:   𝜓(𝑛)

Proof of Theorem zsupcl
Dummy variables 𝑥 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsupcl.m . . . 4 (𝜑𝑀 ∈ ℤ)
21zred 8763 . . 3 (𝜑𝑀 ∈ ℝ)
3 lttri3 7467 . . . . 5 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
43adantl 271 . . . 4 ((𝜑 ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
5 zssre 8652 . . . . 5 ℤ ⊆ ℝ
6 zsupcl.sbm . . . . . 6 (𝑛 = 𝑀 → (𝜓𝜒))
7 zsupcl.mtru . . . . . 6 (𝜑𝜒)
8 zsupcl.dc . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → DECID 𝜓)
9 zsupcl.bnd . . . . . 6 (𝜑 → ∃𝑗 ∈ (ℤ𝑀)∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
101, 6, 7, 8, 9zsupcllemex 10721 . . . . 5 (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11 ssrexv 3070 . . . . 5 (ℤ ⊆ ℝ → (∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
125, 10, 11mpsyl 64 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
134, 12supclti 6599 . . 3 (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℝ)
146elrab 2759 . . . . 5 (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑀 ∈ ℤ ∧ 𝜒))
151, 7, 14sylanbrc 408 . . . 4 (𝜑𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
164, 12supubti 6600 . . . 4 (𝜑 → (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → ¬ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) < 𝑀))
1715, 16mpd 13 . . 3 (𝜑 → ¬ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) < 𝑀)
182, 13, 17nltled 7506 . 2 (𝜑𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ))
195a1i 9 . . . 4 (𝜑 → ℤ ⊆ ℝ)
204, 10, 19supelti 6603 . . 3 (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ)
21 eluz 8926 . . 3 ((𝑀 ∈ ℤ ∧ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ) → (sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ𝑀) ↔ 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < )))
221, 20, 21syl2anc 403 . 2 (𝜑 → (sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ𝑀) ↔ 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < )))
2318, 22mpbird 165 1 (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ𝑀))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 776   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  wss 2984   class class class wbr 3811  cfv 4968  supcsup 6583  cr 7251   < clt 7424  cle 7425  cz 8645  cuz 8913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-cnex 7338  ax-resscn 7339  ax-1cn 7340  ax-1re 7341  ax-icn 7342  ax-addcl 7343  ax-addrcl 7344  ax-mulcl 7345  ax-addcom 7347  ax-addass 7349  ax-distr 7351  ax-i2m1 7352  ax-0lt1 7353  ax-0id 7355  ax-rnegex 7356  ax-cnre 7358  ax-pre-ltirr 7359  ax-pre-ltwlin 7360  ax-pre-lttrn 7361  ax-pre-apti 7362  ax-pre-ltadd 7363
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-fv 4976  df-riota 5546  df-ov 5593  df-oprab 5594  df-mpt2 5595  df-1st 5845  df-2nd 5846  df-sup 6585  df-pnf 7426  df-mnf 7427  df-xr 7428  df-ltxr 7429  df-le 7430  df-sub 7557  df-neg 7558  df-inn 8316  df-n0 8565  df-z 8646  df-uz 8914  df-fz 9319  df-fzo 9443
This theorem is referenced by:  gcdsupcl  10729
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