Theorem chnwrd 32880

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 32879	. . 3 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	simplbi 496	. 2 ⊢ (𝐶 ∈ ( < Chain𝐴) → 𝐶 ∈ Word 𝐴)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2099  ∀wral 3051   ∖ cdif 3943  {csn 4623   class class class wbr 5145  dom cdm 5674  ‘cfv 6546  (class class class)co 7416  0cc0 11149  1c1 11150   − cmin 11485  Word cword 14517  Chaincchn 32877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-dm 5684  df-iota 6498  df-fv 6554  df-chn 32878

This theorem is referenced by:  pfxchn  32882  chnind  32883  chnub  32884  chnlt  32885  fldext2chn  33601