Theorem qdenval

Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014)

	Ref	Expression
Assertion	qdenval	$$⊢A\in \mathbb{Q}\to \mathrm{denom}\left(A\right)=2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq1	$$⊢a=A\to \left(a=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\leftrightarrow A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)$$
2	1	anbi2d	$$⊢a=A\to \left(\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge a=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\leftrightarrow \left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)$$
3	2	riotabidv	$$⊢a=A\to \left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge a=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)=\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)$$
4	3	fveq2d	$$⊢a=A\to 2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge a=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)=2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)$$
5		df-denom	$$⊢\mathrm{denom}=\left(a\in \mathbb{Q}⟼2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge a=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)\right)$$
6		fvex	$$⊢2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)\in \mathrm{V}$$
7	4 5 6	fvmpt	$$⊢A\in \mathbb{Q}\to \mathrm{denom}\left(A\right)=2^{nd}\left(\left(\iota x\in \left(\mathbb{Z}\times \mathbb{N}\right)|\left(1^{st}\left(x\right)\gcd 2^{nd}\left(x\right)=1\wedge A=\frac{1^{st}\left(x\right)}{2^{nd}\left(x\right)}\right)\right)\right)$$