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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 2821 | . 2 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ Word 𝑆 ↔ 𝐴 ∈ Word 𝑆)) | |
2 | df-upword 45583 | . . . 4 ⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))} | |
3 | 2 | eqabri 2877 | . . 3 ⊢ (𝑤 ∈ UpWord 𝑆 ↔ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))) |
4 | 3 | simplbi 498 | . 2 ⊢ (𝑤 ∈ UpWord 𝑆 → 𝑤 ∈ Word 𝑆) |
5 | 1, 4 | vtoclga 3565 | 1 ⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ 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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