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Mirrors > Home > ILE Home > Th. List > Mathboxes > inffz | GIF version |
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
Ref | Expression |
---|---|
inffz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 499 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ) | |
2 | 1 | zred 8922 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℝ) |
3 | simprr 500 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ) | |
4 | 3 | zred 8922 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℝ) |
5 | 2, 4 | lttri3d 7653 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 = 𝑦 ↔ (¬ 𝑥 < 𝑦 ∧ ¬ 𝑦 < 𝑥))) |
6 | eluzel2 9078 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
7 | eluzfz1 9499 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
8 | elfzle1 9495 | . . . 4 ⊢ (𝑧 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑧) | |
9 | 8 | adantl 272 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑧) |
10 | 6 | zred 8922 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
11 | elfzelz 9494 | . . . . 5 ⊢ (𝑧 ∈ (𝑀...𝑁) → 𝑧 ∈ ℤ) | |
12 | 11 | zred 8922 | . . . 4 ⊢ (𝑧 ∈ (𝑀...𝑁) → 𝑧 ∈ ℝ) |
13 | lenlt 7615 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑀 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑀)) | |
14 | 10, 12, 13 | syl2an 284 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝑀 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑀)) |
15 | 9, 14 | mpbid 146 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 ∈ (𝑀...𝑁)) → ¬ 𝑧 < 𝑀) |
16 | 5, 6, 7, 15 | infminti 6776 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 ‘cfv 5028 (class class class)co 5666 infcinf 6732 ℝcr 7403 < clt 7576 ≤ cle 7577 ℤcz 8804 ℤ≥cuz 9073 ...cfz 9478 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-pre-ltirr 7511 ax-pre-apti 7514 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sup 6733 df-inf 6734 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-neg 7710 df-z 8805 df-uz 9074 df-fz 9479 |
This theorem is referenced by: (None) |
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