Theorem chnltm1

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

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

Proof

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