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	$$⊢A\in \mathscr{H}$$
	hvass.2	$$⊢B\in \mathscr{H}$$
	hvass.3	$$⊢C\in \mathscr{H}$$
Assertion	hvsubassi	$$⊢\left(A{-}_{ℎ}B\right){-}_{ℎ}C=A{-}_{ℎ}\left(B{+}_{ℎ}C\right)$$

Proof

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