Theorem infssfzcldc 10618

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 2301	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓})
4		nfcv 2386	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑀...𝑁)
5	4	elrabsf 3084	. . . . . . 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 10377	. . . 4 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
10	8, 9	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11		eqid 2234	. . 3 ⊢ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)} = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}
12		elfzuz 10374	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
13	12	ad2antrl 490	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → 𝑛 ∈ (ℤ≥‘𝑀))
14		elfzle2 10382	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ≤ 𝑁)
15	14	ad2antrl 490	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝜓)) → 𝑛 ≤ 𝑁)
16		simprr 533	. . . . . . . 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 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ∈ (ℤ≥‘𝑀))
19		simprrl 541	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ≤ 𝑁)
20		eluzelz 9881	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓)) → 𝑛 ∈ ℤ)
22		elfzel2 10376	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
23	8, 22	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		eluz 9885	. . . . . . . . . . 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 10372	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀...𝑁) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑛)))
28	18, 26, 27	sylanbrc 417	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))) → 𝑛 ∈ (𝑀...𝑁))
29		simprrr 542	. . . . . . . 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 600	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ (𝑀...𝑁) ∧ 𝜓) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 ≤ 𝑁 ∧ 𝜓))))
32	31	rabbidva2 2799	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓} = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
33	2, 32	eqtrid 2279	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
34	1, 33	eleqtrd 2313	. . 3 ⊢ (𝜑 → 𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
35		elfzelz 10378	. . . . 5 ⊢ (𝑛 ∈ (𝑀...𝐴) → 𝑛 ∈ ℤ)
36		zdcle 9671	. . . . 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 941	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID (𝑛 ≤ 𝑁 ∧ 𝜓))
40	10, 11, 34, 39	infssuzcldc 10617	. 2 ⊢ (𝜑 → inf({𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}, ℝ, < ) ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)})
41	33	infeq1d 7316	. 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) = inf({𝑛 ∈ (ℤ≥‘𝑀) ∣ (𝑛 ≤ 𝑁 ∧ 𝜓)}, ℝ, < ))
42	40, 41, 33	3eltr4d 2318	1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2205  {crab 2526  [wsbc 3045   class class class wbr 4114  ‘cfv 5357  (class class class)co 6058  infcinf 7287  ℝcr 8142   < clt 8324   ≤ cle 8325  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499

This theorem is referenced by:  ballotfilemscl  13191