Theorem chnwrd 32958

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 32957	. . 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 2104  ∀wral 3057   ∖ cdif 3960  {csn 4630   class class class wbr 5149  dom cdm 5683  ‘cfv 6558  (class class class)co 7425  0cc0 11146  1c1 11147   − cmin 11483  Word cword 14538  Chaincchn 32955

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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-dm 5693  df-iota 6510  df-fv 6566  df-chn 32956

This theorem is referenced by:  pfxchn  32960  chnind  32961  chnub  32962  chnlt  32963  fldext2chn  33697