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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > chnwrd | Structured version Visualization version GIF version |
Description: A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025.) |
Ref | Expression |
---|---|
chnwrd.1 | ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴)) |
Ref | Expression |
---|---|
chnwrd | ⊢ (𝜑 → 𝐶 ∈ Word 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chnwrd.1 | . 2 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴)) | |
2 | ischn 32983 | . . 3 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) | |
3 | 2 | simplbi 497 | . 2 ⊢ (𝐶 ∈ ( < Chain𝐴) → 𝐶 ∈ Word 𝐴) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3060 ∖ cdif 3947 {csn 4624 class class class wbr 5141 dom cdm 5683 ‘cfv 6559 (class class class)co 7429 0cc0 11151 1c1 11152 − cmin 11488 Word cword 14548 Chaincchn 32981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-dm 5693 df-iota 6512 df-fv 6567 df-chn 32982 |
This theorem is referenced by: pfxchn 32986 chnind 32987 chnub 32988 chnlt 32989 fldext2chn 33750 |
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