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Mirrors > Home > MPE Home > Th. List > Mathboxes > inffzOLD | Structured version Visualization version GIF version |
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) Obsolete version of inffz 31946 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
inffzOLD | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, ◡ < ) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssre 11585 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
2 | ltso 10319 | . . . . 5 ⊢ < Or ℝ | |
3 | soss 5188 | . . . . 5 ⊢ (ℤ ⊆ ℝ → ( < Or ℝ → < Or ℤ)) | |
4 | 1, 2, 3 | mp2 9 | . . . 4 ⊢ < Or ℤ |
5 | cnvso 5818 | . . . 4 ⊢ ( < Or ℤ ↔ ◡ < Or ℤ) | |
6 | 4, 5 | mpbi 220 | . . 3 ⊢ ◡ < Or ℤ |
7 | 6 | a1i 11 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ◡ < Or ℤ) |
8 | eluzel2 11892 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
9 | eluzfz1 12554 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
10 | elfzle1 12550 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
11 | 10 | adantl 467 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
12 | 8 | zred 11683 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
13 | elfzelz 12548 | . . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
14 | 13 | zred 11683 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
15 | lenlt 10317 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑀 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑀)) | |
16 | 12, 14, 15 | syl2an 575 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑀 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑀)) |
17 | 11, 16 | mpbid 222 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝑥 < 𝑀) |
18 | brcnvg 5441 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑀◡ < 𝑥 ↔ 𝑥 < 𝑀)) | |
19 | 18 | notbid 307 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ 𝑀◡ < 𝑥 ↔ ¬ 𝑥 < 𝑀)) |
20 | 8, 19 | sylan 561 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ 𝑀◡ < 𝑥 ↔ ¬ 𝑥 < 𝑀)) |
21 | 17, 20 | mpbird 247 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝑀◡ < 𝑥) |
22 | 7, 8, 9, 21 | supmax 8528 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, ◡ < ) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ⊆ wss 3721 class class class wbr 4784 Or wor 5169 ◡ccnv 5248 ‘cfv 6031 (class class class)co 6792 supcsup 8501 ℝcr 10136 < clt 10275 ≤ cle 10276 ℤcz 11578 ℤ≥cuz 11887 ...cfz 12532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-pre-lttri 10211 ax-pre-lttrn 10212 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-1st 7314 df-2nd 7315 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-sup 8503 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-neg 10470 df-z 11579 df-uz 11888 df-fz 12533 |
This theorem is referenced by: (None) |
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