Theorem pilem1

Description: Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro, 9-May-2014)

	Ref	Expression
Assertion	pilem1	$$⊢A\in \left({ℝ}^{+}\cap {\sin }^{-1}\left[\left\{0\right\}\right]\right)\leftrightarrow \left(A\in {ℝ}^{+}\wedge \sin A=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		elin	$$⊢A\in \left({ℝ}^{+}\cap {\sin }^{-1}\left[\left\{0\right\}\right]\right)\leftrightarrow \left(A\in {ℝ}^{+}\wedge A\in {\sin }^{-1}\left[\left\{0\right\}\right]\right)$$
2		sinf	$$⊢\sin :ℂ⟶ℂ$$
3		ffn	$$⊢\sin :ℂ⟶ℂ\to \sin Fnℂ$$
4		fniniseg	$$⊢\sin Fnℂ\to \left(A\in {\sin }^{-1}\left[\left\{0\right\}\right]\leftrightarrow \left(A\in ℂ\wedge \sin A=0\right)\right)$$
5	2 3 4	mp2b	$$⊢A\in {\sin }^{-1}\left[\left\{0\right\}\right]\leftrightarrow \left(A\in ℂ\wedge \sin A=0\right)$$
6		rpcn	$$⊢A\in {ℝ}^{+}\to A\in ℂ$$
7	6	biantrurd	$$⊢A\in {ℝ}^{+}\to \left(\sin A=0\leftrightarrow \left(A\in ℂ\wedge \sin A=0\right)\right)$$
8	5 7	bitr4id	$$⊢A\in {ℝ}^{+}\to \left(A\in {\sin }^{-1}\left[\left\{0\right\}\right]\leftrightarrow \sin A=0\right)$$
9	8	pm5.32i	$$⊢\left(A\in {ℝ}^{+}\wedge A\in {\sin }^{-1}\left[\left\{0\right\}\right]\right)\leftrightarrow \left(A\in {ℝ}^{+}\wedge \sin A=0\right)$$
10	1 9	bitri	$$⊢A\in \left({ℝ}^{+}\cap {\sin }^{-1}\left[\left\{0\right\}\right]\right)\leftrightarrow \left(A\in {ℝ}^{+}\wedge \sin A=0\right)$$