Theorem lpadlen1

Description: Length of a left-padded word, in the case the length of the given word W is at least the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023)

	Ref	Expression
Hypotheses	lpadlen.1	⊢ ( 𝜑 → 𝐿 ∈ ℕ0 )
	lpadlen.2	⊢ ( 𝜑 → 𝑊 ∈ Word 𝑆 )
	lpadlen.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
	lpadlen1.1	⊢ ( 𝜑 → 𝐿 ≤ ( ♯ ‘ 𝑊 ) )
Assertion	lpadlen1	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) ) = ( ♯ ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		lpadlen.1	⊢ ( 𝜑 → 𝐿 ∈ ℕ0 )
2		lpadlen.2	⊢ ( 𝜑 → 𝑊 ∈ Word 𝑆 )
3		lpadlen.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
4		lpadlen1.1	⊢ ( 𝜑 → 𝐿 ≤ ( ♯ ‘ 𝑊 ) )
5	1 2 3	lpadval	⊢ ( 𝜑 → ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) = ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) )
6	1 2 3 4	lpadlem3	⊢ ( 𝜑 → ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) = ∅ )
7	6	oveq1d	⊢ ( 𝜑 → ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) = ( ∅ ++ 𝑊 ) )
8		ccatlid	⊢ ( 𝑊 ∈ Word 𝑆 → ( ∅ ++ 𝑊 ) = 𝑊 )
9	2 8	syl	⊢ ( 𝜑 → ( ∅ ++ 𝑊 ) = 𝑊 )
10	5 7 9	3eqtrd	⊢ ( 𝜑 → ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) = 𝑊 )
11	10	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) ) = ( ♯ ‘ 𝑊 ) )