Theorem mulcomnq

Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995) (Revised by Mario Carneiro, 28-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	mulcomnq	$$⊢A{\cdot }_{𝑸}B=B{\cdot }_{𝑸}A$$

Proof

Step	Hyp	Ref	Expression
1		mulcompq	$$⊢A{\cdot }_{\mathrm{𝑝𝑸}}B=B{\cdot }_{\mathrm{𝑝𝑸}}A$$
2	1	fveq2i	$$⊢{/}_{𝑸}\left(A{\cdot }_{\mathrm{𝑝𝑸}}B\right)={/}_{𝑸}\left(B{\cdot }_{\mathrm{𝑝𝑸}}A\right)$$
3		mulpqnq	$$⊢\left(A\in 𝑸\wedge B\in 𝑸\right)\to A{\cdot }_{𝑸}B={/}_{𝑸}\left(A{\cdot }_{\mathrm{𝑝𝑸}}B\right)$$
4		mulpqnq	$$⊢\left(B\in 𝑸\wedge A\in 𝑸\right)\to B{\cdot }_{𝑸}A={/}_{𝑸}\left(B{\cdot }_{\mathrm{𝑝𝑸}}A\right)$$
5	4	ancoms	$$⊢\left(A\in 𝑸\wedge B\in 𝑸\right)\to B{\cdot }_{𝑸}A={/}_{𝑸}\left(B{\cdot }_{\mathrm{𝑝𝑸}}A\right)$$
6	2 3 5	3eqtr4a	$$⊢\left(A\in 𝑸\wedge B\in 𝑸\right)\to A{\cdot }_{𝑸}B=B{\cdot }_{𝑸}A$$
7		mulnqf	$$⊢{\cdot }_{𝑸}:𝑸\times 𝑸⟶𝑸$$
8	7	fdmi	$$⊢\mathrm{dom}{\cdot }_{𝑸}=𝑸\times 𝑸$$
9	8	ndmovcom	$$⊢\neg \left(A\in 𝑸\wedge B\in 𝑸\right)\to A{\cdot }_{𝑸}B=B{\cdot }_{𝑸}A$$
10	6 9	pm2.61i	$$⊢A{\cdot }_{𝑸}B=B{\cdot }_{𝑸}A$$