Theorem zsupcl 10481

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 9592	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
3		lttri3 8249	. . . . 5 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
4	3	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 < 𝑣 ∧ ¬ 𝑣 < 𝑢)))
5		zssre 9476	. . . . 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 10480	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11		ssrexv 3290	. . . . 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 7188	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℝ)
14	6	elrab 2960	. . . . 5 ⊢ (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑀 ∈ ℤ ∧ 𝜒))
15	1, 7, 14	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
16	4, 12	supubti 7189	. . . 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 8290	. 2 ⊢ (𝜑 → 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ))
19	5	a1i 9	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℝ)
20	4, 10, 19	supelti 7192	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ)
21		eluz 9759	. . 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 411	. 2 ⊢ (𝜑 → (sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < )))
23	18, 22	mpbird 167	1 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512   ⊆ wss 3198   class class class wbr 4086  ‘cfv 5324  supcsup 7172  ℝcr 8021   < clt 8204   ≤ cle 8205  ℤcz 9469  ℤ_≥cuz 9745

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368

This theorem is referenced by:  suprzubdc  10486  gcdsupcl  12519