Theorem chnwrd 32972

Description: A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Hypothesis

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))

Assertion

Ref	Expression
chnwrd	⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Proof of Theorem chnwrd

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnwrd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
2		ischn 32971	. . 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)

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973  {csn 4648   class class class wbr 5166  dom cdm 5695  ‘cfv 6568  (class class class)co 7443  0cc0 11178  1c1 11179   − cmin 11514  Word cword 14556  Chaincchn 32969

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-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5705  df-iota 6520  df-fv 6576  df-chn 32970

This theorem is referenced by:  pfxchn  32974  chnind  32975  chnub  32976  chnlt  32977  fldext2chn  33711