This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem honegdi

Description: Distribution of negative over addition. (Contributed by NM, 24-Aug-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	honegdi	$⊢ (T : ℋ ⟶ ℋ \land U : ℋ ⟶ ℋ) \to -1 \cdot_{op} (T +_{op} U) = (-1 \cdot_{op} T) +_{op} (-1 \cdot_{op} U)$

Step	Hyp	Ref	Expression
1		neg1cn	$⊢ - 1 \in ℂ$
2		hoadddi	$⊢ (- 1 \in ℂ \land T : ℋ ⟶ ℋ \land U : ℋ ⟶ ℋ) \to -1 \cdot_{op} (T +_{op} U) = (-1 \cdot_{op} T) +_{op} (-1 \cdot_{op} U)$
3	1 2	mp3an1	$⊢ (T : ℋ ⟶ ℋ \land U : ℋ ⟶ ℋ) \to -1 \cdot_{op} (T +_{op} U) = (-1 \cdot_{op} T) +_{op} (-1 \cdot_{op} U)$