Theorem zsupcl 11370

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 nonempty 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	⊢ (𝜑 → 𝑀 ∈ ℤ)
zsupcl.sbm	⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
zsupcl.mtru	⊢ (𝜑 → 𝜒)
zsupcl.dc	⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)
zsupcl.bnd	⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)

Assertion

Ref	Expression
zsupcl	⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀))

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.

Step	Hyp	Ref	Expression
1		zsupcl.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2	1	zred 8967	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
3		lttri3 7662	. . . . 5 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
4	3	adantl 272	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
5		zssre 8855	. . . . 5 ⊢ ℤ ⊆ ℝ
6		zsupcl.sbm	. . . . . 6 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
7		zsupcl.mtru	. . . . . 6 ⊢ (𝜑 → 𝜒)
8		zsupcl.dc	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)
9		zsupcl.bnd	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)
10	1, 6, 7, 8, 9	zsupcllemex 11369	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11		ssrexv 3101	. . . . 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))))
12	5, 10, 11	mpsyl 65	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
13	4, 12	supclti 6773	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℝ)
14	6	elrab 2785	. . . . 5 ⊢ (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑀 ∈ ℤ ∧ 𝜒))
15	1, 7, 14	sylanbrc 409	. . . 4 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
16	4, 12	supubti 6774	. . . 4 ⊢ (𝜑 → (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → ¬ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) < 𝑀))
17	15, 16	mpd 13	. . 3 ⊢ (𝜑 → ¬ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) < 𝑀)
18	2, 13, 17	nltled 7701	. 2 ⊢ (𝜑 → 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ))
19	5	a1i 9	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℝ)
20	4, 10, 19	supelti 6777	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ)
21		eluz 9131	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ) → (sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < )))
22	1, 20, 21	syl2anc 404	. 2 ⊢ (𝜑 → (sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < )))
23	18, 22	mpbird 166	1 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 783   = wceq 1296   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  {crab 2374   ⊆ wss 3013   class class class wbr 3867  ‘cfv 5049  supcsup 6757  ℝcr 7446   < clt 7619   ≤ cle 7620  ℤcz 8848  ℤ_≥cuz 9118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-fz 9574  df-fzo 9703

This theorem is referenced by:  gcdsupcl  11377