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Metamath Proof Explorer

Theorem pfxcctswrd

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

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

Proof

Step Hyp Ref Expression
1 lencl ⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 )
2 nn0fz0 ⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ↔ ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
3 1 2 sylib ⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
4 3 adantr ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) )
5 ccatpfx ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 prefix 𝑀 ) ++ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( 𝑊 prefix ( ♯ ‘ 𝑊 ) ) )
6 4 5 mpd3an3 ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 prefix 𝑀 ) ++ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = ( 𝑊 prefix ( ♯ ‘ 𝑊 ) ) )
7 pfxid ⊢ ( 𝑊 ∈ Word 𝑉 → ( 𝑊 prefix ( ♯ ‘ 𝑊 ) ) = 𝑊 )
8 7 adantr ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 prefix ( ♯ ‘ 𝑊 ) ) = 𝑊 )
9 6 8 eqtrd ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 prefix 𝑀 ) ++ ( 𝑊 substr ⟨ 𝑀 , ( ♯ ‘ 𝑊 ) ⟩ ) ) = 𝑊 )