Definition df-pi 15175

Description: Define the constant pi, π = 3.14159..., which is the smallest positive number whose sine is zero. Definition of π in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)

Assertion

Ref	Expression
df-pi	⊢ π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )

Detailed syntax breakdown of Definition df-pi

Step	Hyp	Ref	Expression
1		cpi 15169	. 2 class π
2		crp 12112	. . . 4 class ℝ+
3		csin 15166	. . . . . 6 class sin
4	3	ccnv 5341	. . . . 5 class ◡sin
5		cc0 10252	. . . . . 6 class 0
6	5	csn 4397	. . . . 5 class {0}
7	4, 6	cima 5345	. . . 4 class (◡sin “ {0})
8	2, 7	cin 3797	. . 3 class (ℝ+ ∩ (◡sin “ {0}))
9		cr 10251	. . 3 class ℝ
10		clt 10391	. . 3 class <
11	8, 9, 10	cinf 8616	. 2 class inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )
12	1, 11	wceq 1658	1 wff π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )

This definition is referenced by:  pilem2  24605  pilem3  24606  pilem3OLD  24607