Theorem infssfzcldc 10687

Description: The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)

Hypotheses

Ref	Expression
infssfzledc.s	⊢ 𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}
infssfzledc.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
infssfzledc.dc	⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)

Assertion

Ref	Expression
infssfzcldc	⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)

Distinct variable groups:   𝐴,𝑛   𝑛,𝑀   𝑛,𝑁   𝜑,𝑛

Allowed substitution hints:   𝜓(𝑛)   𝑆(𝑛)

Proof of Theorem infssfzcldc

Step	Hyp	Ref	Expression
1		infssfzledc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
2		infssfzledc.s	. . . . . . . 8 ⊢ 𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}
3	2	eleq2i 2305	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓})
4		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑀...𝑁)
5	4	elrabsf 3090	. . . . . . 7 ⊢ (𝐴 ∈ {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓} ↔ (𝐴 ∈ (𝑀...𝑁) ∧ [𝐴 / 𝑛]𝜓))
6	3, 5	bitri 184	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ (𝑀...𝑁) ∧ [𝐴 / 𝑛]𝜓))
7	1, 6	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (𝑀...𝑁) ∧ [𝐴 / 𝑛]𝜓))
8	7	simpld 112	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁))
9		elfzel1 10445	. . . 4 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
10	8, 9	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11		eqid 2238	. . 3 ⊢ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)} = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}
12		elfzuz 10442	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
13	12	ad2antrl 494	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → 𝑛 ∈ (ℤ≥‘𝑀))
14		elfzle2 10450	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ≤ 𝑁)
15	14	ad2antrl 494	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → 𝑛 ≤ 𝑁)
16		simprr 537	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → 𝜓)
17	13, 15, 16	jca32 310	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓)))
18		simprl 535	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ∈ (ℤ≥‘𝑀))
19		simprrl 545	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ≤ 𝑁)
20		eluzelz 9948	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓)) → 𝑛 ∈ ℤ)
22		elfzel2 10444	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
23	8, 22	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		eluz 9952	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑛) ↔ 𝑛 ≤ 𝑁))
25	21, 23, 24	syl2anr 290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → (𝑁 ∈ (ℤ≥‘𝑛) ↔ 𝑛 ≤ 𝑁))
26	19, 25	mpbird 167	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑁 ∈ (ℤ≥‘𝑛))
27		elfzuzb 10440	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑛)))
28	18, 26, 27	sylanbrc 421	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ∈ (𝑀...𝑁))
29		simprrr 546	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝜓)
30	28, 29	jca 306	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓))
31	17, 30	impbida 604	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ (𝑀...𝑁) ∧ 𝜓) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))))
32	31	rabbidva2 2805	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓} = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
33	2, 32	eqtrid 2283	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
34	1, 33	eleqtrd 2317	. . 3 ⊢ (𝜑 → 𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
35		elfzelz 10446	. . . . 5 ⊢ (𝑛 ∈ (𝑀...𝐴) → 𝑛 ∈ ℤ)
36		zdcle 9733	. . . . 5 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑛 ≤ 𝑁)
37	35, 23, 36	syl2anr 290	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝑛 ≤ 𝑁)
38		infssfzledc.dc	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)
39	37, 38	dcand 945	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID (𝑛 ≤ 𝑁 ∧ 𝜓))
40	10, 11, 34, 39	infssuzcldc 10686	. 2 ⊢ (𝜑 → inf({𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}, ℝ, < ) ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
41	33	infeq1d 7359	. 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) = inf({𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}, ℝ, < ))
42	40, 41, 33	3eltr4d 2322	1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {crab 2532  [wsbc 3051   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  infcinf 7330  ℝcr 8185   < clt 8367   ≤ cle 8368  ℤcz 9656  ℤ_≥cuz 9938  ...cfz 10429

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-fzo 10568

This theorem is used by:  ballotfilemscl  13306