Theorem chnltm1 18833

Description: Basic property of a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Hypotheses

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
chnltm1.2	⊢ (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))

Assertion

Ref	Expression
chnltm1	⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Proof of Theorem chnltm1

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 7474	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘(𝑛 − 1)) = (𝐶‘(𝑁 − 1)))
2		fveq2 6916	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘𝑛) = (𝐶‘𝑁))
3	1, 2	breq12d 5121	. 2 ⊢ (𝑛 = 𝑁 → ((𝐶‘(𝑛 − 1)) < (𝐶‘𝑛) ↔ (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁)))
4		chnwrd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
5		ischn 18831	. . . 4 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	4, 5	sylib 221	. . 3 ⊢ (𝜑 → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7	6	simprd 501	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))
8		chnltm1.2	. 2 ⊢ (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))
9	3, 7, 8	rspcdva 3581	1 ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∖ cdif 4048  {csn 4588   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  (class class class)co 7451  0cc0 11239  1c1 11240   − cmin 11580  Word cword 14699   Chain cchn 18829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-ov 7454  df-chn 18830

This theorem is used by:  pfxchn  18834  chnccat  18850  chnrev  18851  chnsubseq  48163