Theorem lenrevpfxcctswrd

Description: The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018) (Revised by AV, 9-May-2020)

	Ref	Expression
Assertion	lenrevpfxcctswrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ++ ( 𝑊 prefix 𝑀 ) ) ) = ( ♯ ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		swrdcl	⊢ ( 𝑊 ∈ Word 𝑉 → ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ∈ Word 𝑉 )
2		pfxcl	⊢ ( 𝑊 ∈ Word 𝑉 → ( 𝑊 prefix 𝑀 ) ∈ Word 𝑉 )
3		ccatlen	⊢ ( ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ∈ Word 𝑉 ∧ ( 𝑊 prefix 𝑀 ) ∈ Word 𝑉 ) → ( ♯ ‘ ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ++ ( 𝑊 prefix 𝑀 ) ) ) = ( ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) + ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ) )
4	1 2 3	syl2anc	⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ++ ( 𝑊 prefix 𝑀 ) ) ) = ( ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) + ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ) )
5	4	adantr	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ++ ( 𝑊 prefix 𝑀 ) ) ) = ( ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) + ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ) )
6		swrdrlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( ( ♯ ‘ 𝑊 ) − 𝑀 ) )
7		fznn0sub	⊢ ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 𝑀 ) ∈ ℕ0 )
8	7	adantl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ 𝑊 ) − 𝑀 ) ∈ ℕ0 )
9	6 8	eqeltrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ∈ ℕ0 )
10	9	nn0cnd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ∈ ℂ )
11		pfxlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) = 𝑀 )
12		elfznn0	⊢ ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → 𝑀 ∈ ℕ0 )
13	12	adantl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑀 ∈ ℕ0 )
14	11 13	eqeltrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ∈ ℕ0 )
15	14	nn0cnd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ∈ ℂ )
16	10 15	addcomd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) + ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) ) = ( ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) + ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ) )
17		addlenpfx	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) + ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ) = ( ♯ ‘ 𝑊 ) )
18	5 16 17	3eqtrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ++ ( 𝑊 prefix 𝑀 ) ) ) = ( ♯ ‘ 𝑊 ) )