Theorem zsupcl 11950

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 9377	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
3		lttri3 8039	. . . . 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 9262	. . . . 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 11949	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11		ssrexv 3222	. . . . 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 6999	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℝ)
14	6	elrab 2895	. . . . 5 ⊢ (𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑀 ∈ ℤ ∧ 𝜒))
15	1, 7, 14	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
16	4, 12	supubti 7000	. . . 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 8080	. 2 ⊢ (𝜑 → 𝑀 ≤ sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ))
19	5	a1i 9	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℝ)
20	4, 10, 19	supelti 7003	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ ℤ)
21		eluz 9543	. . 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 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3131   class class class wbr 4005  ‘cfv 5218  supcsup 6983  ℝcr 7812   < clt 7994   ≤ cle 7995  ℤcz 9255  ℤ_≥cuz 9530

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145

This theorem is referenced by:  suprzubdc  11955  gcdsupcl  11961