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

Metamath Proof Explorer

Theorem hvsubassi

Description: Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	$⊢ A \in ℋ$
	hvass.2	$⊢ B \in ℋ$
	hvass.3	$⊢ C \in ℋ$
Assertion	hvsubassi	$⊢ (A -_{ℎ} B) -_{ℎ} C = A -_{ℎ} (B +_{ℎ} C)$

Proof

Step	Hyp	Ref	Expression
1		hvass.1	$⊢ A \in ℋ$
2		hvass.2	$⊢ B \in ℋ$
3		hvass.3	$⊢ C \in ℋ$
4		hvsubass	$⊢ (A \in ℋ \land B \in ℋ \land C \in ℋ) \to (A -_{ℎ} B) -_{ℎ} C = A -_{ℎ} (B +_{ℎ} C)$
5	1 2 3 4	mp3an	$⊢ (A -_{ℎ} B) -_{ℎ} C = A -_{ℎ} (B +_{ℎ} C)$