Theorem zsupcl 10490

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 9601	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
3		lttri3 8258	. . . . 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 9485	. . . . 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 10489	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11		ssrexv 3292	. . . . 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 7196	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℝ)
14	6	elrab 2962	. . . . 5 ⊢ (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑀 ∈ ℤ ∧ 𝜒))
15	1, 7, 14	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
16	4, 12	supubti 7197	. . . 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 8299	. 2 ⊢ (𝜑 → 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ))
19	5	a1i 9	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℝ)
20	4, 10, 19	supelti 7200	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ)
21		eluz 9768	. . 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514   ⊆ wss 3200   class class class wbr 4088  ‘cfv 5326  supcsup 7180  ℝcr 8030   < clt 8213   ≤ cle 8214  ℤcz 9478  ℤ_≥cuz 9754

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377

This theorem is referenced by:  suprzubdc  10495  gcdsupcl  12528