Theorem addcomgi

Description: Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012) (Revised by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	addcomgi	$$⊢A+B=B+A$$

Proof

Step	Hyp	Ref	Expression
1		addcom	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+B=B+A$$
2		ax-addf	$$⊢+:ℂ\times ℂ⟶ℂ$$
3	2	fdmi	$$⊢\mathrm{dom}+=ℂ\times ℂ$$
4	3	ndmovcom	$$⊢\neg \left(A\in ℂ\wedge B\in ℂ\right)\to A+B=B+A$$
5	1 4	pm2.61i	$$⊢A+B=B+A$$