Theorem chnpolfz

Description: Provided that chain's relation is a partial order, the chain length is restricted to a specific integer range. (Contributed by Ender Ting, 20-Jan-2026)

	Ref	Expression
Hypotheses	chnpolfz.1	⊢ ( 𝜑 → < Po 𝐴 )
	chnpolfz.2	⊢ ( 𝜑 → 𝐵 ∈ ( < Chain 𝐴 ) )
	chnpolfz.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
Assertion	chnpolfz	⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		chnpolfz.1	⊢ ( 𝜑 → < Po 𝐴 )
2		chnpolfz.2	⊢ ( 𝜑 → 𝐵 ∈ ( < Chain 𝐴 ) )
3		chnpolfz.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
4		0zd	⊢ ( 𝜑 → 0 ∈ ℤ )
5		hashcl	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
6	3 5	syl	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
7	6	nn0zd	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℤ )
8	2	chnwrd	⊢ ( 𝜑 → 𝐵 ∈ Word 𝐴 )
9		lencl	⊢ ( 𝐵 ∈ Word 𝐴 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 )
10	8 9	syl	⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 )
11	10	nn0zd	⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℤ )
12		hashge0	⊢ ( 𝐵 ∈ ( < Chain 𝐴 ) → 0 ≤ ( ♯ ‘ 𝐵 ) )
13	2 12	syl	⊢ ( 𝜑 → 0 ≤ ( ♯ ‘ 𝐵 ) )
14	1 2 3	chnpolleha	⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ≤ ( ♯ ‘ 𝐴 ) )
15	4 7 11 13 14	elfzd	⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) )