Users' Mathboxes Mathbox for Ender Ting < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-upword Structured version   Visualization version   GIF version

Definition df-upword 45891
Description: Strictly increasing sequence is a sequence, adjacent elements of which increase. (Contributed by Ender Ting, 19-Nov-2024.)
Assertion
Ref Expression
df-upword UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1)))}
Distinct variable group:   𝑆,𝑘,𝑤

Detailed syntax breakdown of Definition df-upword
StepHypRef Expression
1 cS . . 3 class 𝑆
21cupword 45890 . 2 class UpWord 𝑆
3 vw . . . . . 6 setvar 𝑤
43cv 1538 . . . . 5 class 𝑤
51cword 14468 . . . . 5 class Word 𝑆
64, 5wcel 2104 . . . 4 wff 𝑤 ∈ Word 𝑆
7 vk . . . . . . . 8 setvar 𝑘
87cv 1538 . . . . . . 7 class 𝑘
98, 4cfv 6542 . . . . . 6 class (𝑤𝑘)
10 c1 11113 . . . . . . . 8 class 1
11 caddc 11115 . . . . . . . 8 class +
128, 10, 11co 7411 . . . . . . 7 class (𝑘 + 1)
1312, 4cfv 6542 . . . . . 6 class (𝑤‘(𝑘 + 1))
14 clt 11252 . . . . . 6 class <
159, 13, 14wbr 5147 . . . . 5 wff (𝑤𝑘) < (𝑤‘(𝑘 + 1))
16 cc0 11112 . . . . . 6 class 0
17 chash 14294 . . . . . . . 8 class
184, 17cfv 6542 . . . . . . 7 class (♯‘𝑤)
19 cmin 11448 . . . . . . 7 class
2018, 10, 19co 7411 . . . . . 6 class ((♯‘𝑤) − 1)
21 cfzo 13631 . . . . . 6 class ..^
2216, 20, 21co 7411 . . . . 5 class (0..^((♯‘𝑤) − 1))
2315, 7, 22wral 3059 . . . 4 wff 𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1))
246, 23wa 394 . . 3 wff (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1)))
2524, 3cab 2707 . 2 class {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1)))}
262, 25wceq 1539 1 wff UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1)))}
Colors of variables: wff setvar class
This definition is referenced by:  upwordnul  45892  upwordisword  45893  upwordsing  45896  tworepnotupword  45898
  Copyright terms: Public domain W3C validator