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Mirrors > Home > MPE Home > Th. List > Mathboxes > upwordisword | Structured version Visualization version GIF version |
Description: Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.) |
Ref | Expression |
---|---|
upwordisword | ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2832 | . 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆)) | |
2 | df-upword 46800 | . . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))} | |
3 | 2 | eqabri 2888 | . . 3 ⊢ (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))) |
4 | 3 | simplbi 497 | . 2 ⊢ (𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆) |
5 | 1, 4 | vtoclga 3589 | 1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 ‘cfv 6575 (class class class)co 7450 0cc0 11186 1c1 11187 + caddc 11189 < clt 11326 − cmin 11522 ..^cfzo 13713 ♯chash 14381 Word cword 14564 UpWord cupword 46799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2178 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-upword 46800 |
This theorem is referenced by: singoutnupword 46804 upwordsseti 46806 upwrdfi 46808 |
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