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Database REAL AND COMPLEX NUMBERS Words over a set Subwords and concatenations 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 ) )