Theorem hvsub32

Description: Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	hvsub32	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{-}_{ℎ}B\right){-}_{ℎ}C=\left(A{-}_{ℎ}C\right){-}_{ℎ}B$$

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	$$⊢\left(B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to B{+}_{ℎ}C=C{+}_{ℎ}B$$
2	1	3adant1	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to B{+}_{ℎ}C=C{+}_{ℎ}B$$
3	2	oveq2d	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to A{-}_{ℎ}\left(B{+}_{ℎ}C\right)=A{-}_{ℎ}\left(C{+}_{ℎ}B\right)$$
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		hvsubass	$$⊢\left(A\in \mathscr{H}\wedge C\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{-}_{ℎ}C\right){-}_{ℎ}B=A{-}_{ℎ}\left(C{+}_{ℎ}B\right)$$
6	5	3com23	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{-}_{ℎ}C\right){-}_{ℎ}B=A{-}_{ℎ}\left(C{+}_{ℎ}B\right)$$
7	3 4 6	3eqtr4d	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{-}_{ℎ}B\right){-}_{ℎ}C=\left(A{-}_{ℎ}C\right){-}_{ℎ}B$$