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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 31614 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 11384 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
2 | ltso 10118 | . . . . 5 ⊢ < Or ℝ | |
3 | soss 5053 | . . . . 5 ⊢ (ℤ ⊆ ℝ → ( < Or ℝ → < Or ℤ)) | |
4 | 1, 2, 3 | mp2 9 | . . . 4 ⊢ < Or ℤ |
5 | cnvso 5674 | . . . 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 11692 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
9 | eluzfz1 12348 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
10 | elfzle1 12344 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
11 | 10 | adantl 482 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
12 | 8 | zred 11482 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
13 | elfzelz 12342 | . . . . . 6 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
14 | 13 | zred 11482 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
15 | lenlt 10116 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑀 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑀)) | |
16 | 12, 14, 15 | syl2an 494 | . . . 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 5303 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑀◡ < 𝑥 ↔ 𝑥 < 𝑀)) | |
19 | 18 | notbid 308 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (𝑀...𝑁)) → (¬ 𝑀◡ < 𝑥 ↔ ¬ 𝑥 < 𝑀)) |
20 | 8, 19 | sylan 488 | . . 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 8373 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, ◡ < ) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 Or wor 5034 ◡ccnv 5113 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℝcr 9935 < clt 10074 ≤ cle 10075 ℤcz 11377 ℤ≥cuz 11687 ...cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: (None) |
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