Theorem hvadd32

Description: Commutative/associative law. (Contributed by NM, 16-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	hvadd32	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}C=\left(A{+}_{ℎ}C\right){+}_{ℎ}B$$

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	oveq2d	$$⊢\left(B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to A{+}_{ℎ}\left(B{+}_{ℎ}C\right)=A{+}_{ℎ}\left(C{+}_{ℎ}B\right)$$
3	2	3adant1	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to A{+}_{ℎ}\left(B{+}_{ℎ}C\right)=A{+}_{ℎ}\left(C{+}_{ℎ}B\right)$$
4		ax-hvass	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}C=A{+}_{ℎ}\left(B{+}_{ℎ}C\right)$$
5		ax-hvass	$$⊢\left(A\in \mathscr{H}\wedge C\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}C\right){+}_{ℎ}B=A{+}_{ℎ}\left(C{+}_{ℎ}B\right)$$
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	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}C=\left(A{+}_{ℎ}C\right){+}_{ℎ}B$$