Theorem swrdrlen

Description: Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018)

	Ref	Expression
Assertion	swrdrlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝐼 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( ( ♯ ‘ 𝑊 ) − 𝐼 ) )

Proof

Step	Hyp	Ref	Expression
1		lencl	⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
2		nn0fz0	⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ↔ ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
3	1 2	sylib	⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
4	3	adantr	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
5		swrdlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝐼 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( ( ♯ ‘ 𝑊 ) − 𝐼 ) )
6	4 5	mpd3an3	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝐼 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( ( ♯ ‘ 𝑊 ) − 𝐼 ) )