Theorem hvadd12

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	hvadd12	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to A{+}_{ℎ}\left(B{+}_{ℎ}C\right)=B{+}_{ℎ}\left(A{+}_{ℎ}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{+}_{ℎ}B=B{+}_{ℎ}A$$
2	1	oveq1d	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}C=\left(B{+}_{ℎ}A\right){+}_{ℎ}C$$
3	2	3adant3	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}C=\left(B{+}_{ℎ}A\right){+}_{ℎ}C$$
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	$$⊢\left(B\in \mathscr{H}\wedge A\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(B{+}_{ℎ}A\right){+}_{ℎ}C=B{+}_{ℎ}\left(A{+}_{ℎ}C\right)$$
6	5	3com12	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(B{+}_{ℎ}A\right){+}_{ℎ}C=B{+}_{ℎ}\left(A{+}_{ℎ}C\right)$$
7	3 4 6	3eqtr3d	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to A{+}_{ℎ}\left(B{+}_{ℎ}C\right)=B{+}_{ℎ}\left(A{+}_{ℎ}C\right)$$