Theorem pfxnd

Description: The value of a prefix operation for a length argument larger than the word length is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv ). (Contributed by AV, 3-May-2020)

	Ref	Expression
Assertion	pfxnd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 prefix 𝐿 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		pfxval	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ) → ( 𝑊 prefix 𝐿 ) = ( 𝑊 substr ⟨ 0 , 𝐿 ⟩ ) )
2	1	3adant3	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 prefix 𝐿 ) = ( 𝑊 substr ⟨ 0 , 𝐿 ⟩ ) )
3		simp1	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → 𝑊 ∈ Word 𝑉 )
4		0zd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → 0 ∈ ℤ )
5		nn0z	⊢ ( 𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ )
6	5	3ad2ant2	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → 𝐿 ∈ ℤ )
7	3 4 6	3jca	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ ∧ 𝐿 ∈ ℤ ) )
8		3mix3	⊢ ( ( ♯ ‘ 𝑊 ) < 𝐿 → ( 0 < 0 ∨ 𝐿 ≤ 0 ∨ ( ♯ ‘ 𝑊 ) < 𝐿 ) )
9	8	3ad2ant3	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 0 < 0 ∨ 𝐿 ≤ 0 ∨ ( ♯ ‘ 𝑊 ) < 𝐿 ) )
10		swrdnd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ( 0 < 0 ∨ 𝐿 ≤ 0 ∨ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 substr ⟨ 0 , 𝐿 ⟩ ) = ∅ ) )
11	7 9 10	sylc	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 substr ⟨ 0 , 𝐿 ⟩ ) = ∅ )
12	2 11	eqtrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) < 𝐿 ) → ( 𝑊 prefix 𝐿 ) = ∅ )