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Theorem upwordisword 45585
Description: Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.)
Assertion
Ref Expression
upwordisword (𝐴 ∈ UpWord 𝑆𝐴 ∈ Word 𝑆)

Proof of Theorem upwordisword
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2821 . 2 (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆𝐴 ∈ Word 𝑆))
2 df-upword 45583 . . . 4 UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1)))}
32eqabri 2877 . . 3 (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤𝑘) < (𝑤‘(𝑘 + 1))))
43simplbi 498 . 2 (𝑤 ∈ UpWord 𝑆𝑤 ∈ Word 𝑆)
51, 4vtoclga 3565 1 (𝐴 ∈ UpWord 𝑆𝐴 ∈ Word 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3061   class class class wbr 5148  cfv 6543  (class class class)co 7408  0cc0 11109  1c1 11110   + caddc 11112   < clt 11247  cmin 11443  ..^cfzo 13626  chash 14289  Word cword 14463  UpWord cupword 45582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-upword 45583
This theorem is referenced by:  singoutnupword  45587  upwordsseti  45589  upwrdfi  45591
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