Theorem chnltm1 32878

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

Hypotheses

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
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 7436	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘(𝑛 − 1)) = (𝐶‘(𝑁 − 1)))
2		fveq2 6890	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘𝑛) = (𝐶‘𝑁))
3	1, 2	breq12d 5156	. 2 ⊢ (𝑛 = 𝑁 → ((𝐶‘(𝑛 − 1)) < (𝐶‘𝑛) ↔ (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁)))
4		chnwrd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
5		ischn 32876	. . . 4 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	4, 5	sylib 217	. . 3 ⊢ (𝜑 → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7	6	simprd 494	. 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 3608	1 ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051   ∖ cdif 3943  {csn 4623   class class class wbr 5143  dom cdm 5672  ‘cfv 6543  (class class class)co 7413  0cc0 11146  1c1 11147   − cmin 11482  Word cword 14514  Chaincchn 32874

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 5144  df-dm 5682  df-iota 6495  df-fv 6551  df-ov 7416  df-chn 32875

This theorem is referenced by:  pfxchn  32879