Theorem chnwrd 18568

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 18567	. . 3 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	simplbi 496	. 2 ⊢ (𝐶 ∈ ( < Chain 𝐴) → 𝐶 ∈ Word 𝐴)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3887  {csn 4568   class class class wbr 5086  dom cdm 5625  ‘cfv 6493  (class class class)co 7361  0cc0 11032  1c1 11033   − cmin 11371  Word cword 14469   Chain cchn 18565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5635  df-iota 6449  df-fv 6501  df-chn 18566

This theorem is referenced by:  pfxchn  18570  chnexg  18578  chnind  18581  chnub  18582  chnlt  18583  chnccats1  18585  chnccat  18586  chnrev  18587  chnflenfi  18588  chnf  18589  chnpolleha  18592  chnpolfz  18593  fldext2chn  33891  constrextdg2lem  33911  constrext2chnlem  33913  chnsubseqword  47327  chnsubseqwl  47328  chnsubseq  47329  chnsuslle  47330  chnerlem1  47331  chnerlem2  47332  chner  47334