Theorem taupilem3

Description: Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019)

	Ref	Expression
Assertion	taupilem3	$$⊢A\in \left({ℝ}^{+}\cap {\cos }^{-1}\left[\left\{1\right\}\right]\right)\leftrightarrow \left(A\in {ℝ}^{+}\wedge \cos A=1\right)$$

Proof

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