Theorem lenpfxcctswrd

Description: The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018) (Revised by AV, 9-May-2020)

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

Proof

Step	Hyp	Ref	Expression
1		pfxcctswrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 prefix 𝑀 ) ++ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = 𝑊 )
2	1	fveq2d	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( ( 𝑊 prefix 𝑀 ) ++ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) ) = ( ♯ ‘ 𝑊 ) )