Theorem chnwrd 18514

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 ∈ ( < Chain 𝐴))

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 ∈ ( < Chain 𝐴))
2		ischn 18513	. . 3 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	simplbi 497	. 2 ⊢ (𝐶 ∈ ( < Chain 𝐴) → 𝐶 ∈ Word 𝐴)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3894  {csn 4573   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  (class class class)co 7346  0cc0 11006  1c1 11007   − cmin 11344  Word cword 14420   Chain cchn 18511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-dm 5624  df-iota 6437  df-fv 6489  df-chn 18512

This theorem is referenced by:  pfxchn  18516  chnexg  18524  chnind  18527  chnub  18528  chnlt  18529  chnccats1  18531  chnccat  18532  chnrev  18533  chnflenfi  18534  chnf  18535  chnpolleha  18538  chnpolfz  18539  fldext2chn  33741  constrextdg2lem  33761  constrext2chnlem  33763