Theorem addlenpfx

Description: The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018) (Revised by AV, 3-May-2020)

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

Proof

Step	Hyp	Ref	Expression
1		pfxlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) = 𝑀 )
2		swrdrlen	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( ( ♯ ‘ 𝑊 ) − 𝑀 ) )
3	1 2	oveq12d	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) + ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ) = ( 𝑀 + ( ( ♯ ‘ 𝑊 ) − 𝑀 ) ) )
4		elfznn0	⊢ ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → 𝑀 ∈ ℕ0 )
5	4	nn0cnd	⊢ ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → 𝑀 ∈ ℂ )
6		lencl	⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
7	6	nn0cnd	⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℂ )
8		pncan3	⊢ ( ( 𝑀 ∈ ℂ ∧ ( ♯ ‘ 𝑊 ) ∈ ℂ ) → ( 𝑀 + ( ( ♯ ‘ 𝑊 ) − 𝑀 ) ) = ( ♯ ‘ 𝑊 ) )
9	5 7 8	syl2anr	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑀 + ( ( ♯ ‘ 𝑊 ) − 𝑀 ) ) = ( ♯ ‘ 𝑊 ) )
10	3 9	eqtrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 prefix 𝑀 ) ) + ( ♯ ‘ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ) = ( ♯ ‘ 𝑊 ) )