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Mirrors > Home > MPE Home > Th. List > Mathboxes > inffz | Structured version Visualization version GIF version |
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
inffz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssre 11976 | . . . 4 ⊢ ℤ ⊆ ℝ | |
2 | ltso 10709 | . . . 4 ⊢ < Or ℝ | |
3 | soss 5486 | . . . 4 ⊢ (ℤ ⊆ ℝ → ( < Or ℝ → < Or ℤ)) | |
4 | 1, 2, 3 | mp2 9 | . . 3 ⊢ < Or ℤ |
5 | 4 | a1i 11 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → < Or ℤ) |
6 | eluzel2 12236 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
7 | eluzfz1 12902 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
8 | elfzle1 12898 | . . . 4 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
9 | 8 | adantl 482 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
10 | 6 | zred 12075 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
11 | elfzelz 12896 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
12 | 11 | zred 12075 | . . . 4 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
13 | lenlt 10707 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑀 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑀)) | |
14 | 10, 12, 13 | syl2an 595 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑀 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑀)) |
15 | 9, 14 | mpbid 233 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝑥 < 𝑀) |
16 | 5, 6, 7, 15 | infmin 8946 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 Or wor 5466 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℝcr 10524 < clt 10663 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-z 11970 df-uz 12232 df-fz 12881 |
This theorem is referenced by: (None) |
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