Theorem mulpipq2

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	mulpipq2	$$⊢\left(A\in \left(𝑵\times 𝑵\right)\wedge B\in \left(𝑵\times 𝑵\right)\right)\to A{\cdot }_{\mathrm{𝑝𝑸}}B=⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(B\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(B\right)⟩$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢x=A\to 1^{st}\left(x\right)=1^{st}\left(A\right)$$
2	1	oveq1d	$$⊢x=A\to 1^{st}\left(x\right){\cdot }_{𝑵}{1}^{st}\left(y\right)=1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(y\right)$$
3		fveq2	$$⊢x=A\to 2^{nd}\left(x\right)=2^{nd}\left(A\right)$$
4	3	oveq1d	$$⊢x=A\to 2^{nd}\left(x\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)=2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)$$
5	2 4	opeq12d	$$⊢x=A\to ⟨1^{st}\left(x\right){\cdot }_{𝑵}{1}^{st}\left(y\right),2^{nd}\left(x\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)⟩=⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(y\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)⟩$$
6		fveq2	$$⊢y=B\to 1^{st}\left(y\right)=1^{st}\left(B\right)$$
7	6	oveq2d	$$⊢y=B\to 1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(y\right)=1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(B\right)$$
8		fveq2	$$⊢y=B\to 2^{nd}\left(y\right)=2^{nd}\left(B\right)$$
9	8	oveq2d	$$⊢y=B\to 2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)=2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(B\right)$$
10	7 9	opeq12d	$$⊢y=B\to ⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(y\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)⟩=⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(B\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(B\right)⟩$$
11		df-mpq	$$⊢{\cdot }_{\mathrm{𝑝𝑸}}=\left(x\in \left(𝑵\times 𝑵\right),y\in \left(𝑵\times 𝑵\right)⟼⟨1^{st}\left(x\right){\cdot }_{𝑵}{1}^{st}\left(y\right),2^{nd}\left(x\right){\cdot }_{𝑵}{2}^{nd}\left(y\right)⟩\right)$$
12		opex	$$⊢⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(B\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(B\right)⟩\in \mathrm{V}$$
13	5 10 11 12	ovmpo	$$⊢\left(A\in \left(𝑵\times 𝑵\right)\wedge B\in \left(𝑵\times 𝑵\right)\right)\to A{\cdot }_{\mathrm{𝑝𝑸}}B=⟨1^{st}\left(A\right){\cdot }_{𝑵}{1}^{st}\left(B\right),2^{nd}\left(A\right){\cdot }_{𝑵}{2}^{nd}\left(B\right)⟩$$