Metamath Proof Explorer

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	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to W prefix L = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		pfxval	$⊢ (W \in Word V \land L \in ℕ_{0}) \to W prefix L = W substr ⟨0, L⟩$
2	1	3adant3	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to W prefix L = W substr ⟨0, L⟩$
3		simp1	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to W \in Word V$
4		0zd	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to 0 \in ℤ$
5		nn0z	$⊢ L \in ℕ_{0} \to L \in ℤ$
6	5	3ad2ant2	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to L \in ℤ$
7	3 4 6	3jca	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to (W \in Word V \land 0 \in ℤ \land L \in ℤ)$
8		3mix3	$⊢ \|W\| < L \to (0 < 0 \lor L \leq 0 \lor \|W\| < L)$
9	8	3ad2ant3	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to (0 < 0 \lor L \leq 0 \lor \|W\| < L)$
10		swrdnd	$⊢ (W \in Word V \land 0 \in ℤ \land L \in ℤ) \to ((0 < 0 \lor L \leq 0 \lor \|W\| < L) \to W substr ⟨0, L⟩ = \emptyset)$
11	7 9 10	sylc	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to W substr ⟨0, L⟩ = \emptyset$
12	2 11	eqtrd	$⊢ (W \in Word V \land L \in ℕ_{0} \land \|W\| < L) \to W prefix L = \emptyset$