Theorem lpadlen2

Description: Length of a left-padded word, in the case the given word W is shorter than 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 )
	lpadlen2.1	⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ≤ 𝐿 )
Assertion	lpadlen2	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) ) = 𝐿 )

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		lpadlen2.1	⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ≤ 𝐿 )
5	1 2 3	lpadval	⊢ ( 𝜑 → ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) = ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) )
6	5	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) ) = ( ♯ ‘ ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) ) )
7	3	lpadlem1	⊢ ( 𝜑 → ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ∈ Word 𝑆 )
8		ccatlen	⊢ ( ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑆 ) → ( ♯ ‘ ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) ) = ( ( ♯ ‘ ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ) + ( ♯ ‘ 𝑊 ) ) )
9	7 2 8	syl2anc	⊢ ( 𝜑 → ( ♯ ‘ ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) ) = ( ( ♯ ‘ ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ) + ( ♯ ‘ 𝑊 ) ) )
10	1 2 3 4	lpadlem2	⊢ ( 𝜑 → ( ♯ ‘ ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ) = ( 𝐿 − ( ♯ ‘ 𝑊 ) ) )
11	10	oveq1d	⊢ ( 𝜑 → ( ( ♯ ‘ ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ) + ( ♯ ‘ 𝑊 ) ) = ( ( 𝐿 − ( ♯ ‘ 𝑊 ) ) + ( ♯ ‘ 𝑊 ) ) )
12	1	nn0cnd	⊢ ( 𝜑 → 𝐿 ∈ ℂ )
13		lencl	⊢ ( 𝑊 ∈ Word 𝑆 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
14	2 13	syl	⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
15	14	nn0cnd	⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ℂ )
16	12 15	npcand	⊢ ( 𝜑 → ( ( 𝐿 − ( ♯ ‘ 𝑊 ) ) + ( ♯ ‘ 𝑊 ) ) = 𝐿 )
17	9 11 16	3eqtrd	⊢ ( 𝜑 → ( ♯ ‘ ( ( ( 0 ..^ ( 𝐿 − ( ♯ ‘ 𝑊 ) ) ) × { 𝐶 } ) ++ 𝑊 ) ) = 𝐿 )
18	6 17	eqtrd	⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐶 leftpad 𝑊 ) ‘ 𝐿 ) ) = 𝐿 )