Definition df-pi 13810

Description: Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (We use the inverse of less-than, " `' <", to turn supremum into infimum; currently we don't have infimum defined separately.) (Contributed by Paul Chapman, 23-Jan-2008.)

Assertion

Ref	Expression
df-pi	|- pi = sup ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , `' < )

Detailed syntax breakdown of Definition df-pi

Step	Hyp	Ref	Expression
1		cpi 13804	. 2 class pi
2		crp 11139	. . . 4 class RR+
3		csin 13801	. . . . . 6 class sin
4	3	ccnv 4912	. . . . 5 class `' sin
5		cc0 9403	. . . . . 6 class 0
6	5	csn 3944	. . . . 5 class { 0 }
7	4, 6	cima 4916	. . . 4 class ( `' sin " { 0 } )
8	2, 7	cin 3388	. . 3 class ( RR+ i^i ( `' sin " { 0 } ) )
9		cr 9402	. . 3 class RR
10		clt 9539	. . . 4 class <
11	10	ccnv 4912	. . 3 class `' <
12	8, 9, 11	csup 7815	. 2 class sup ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , `' < )
13	1, 12	wceq 1399	1 wff pi = sup ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , `' < )

This definition is referenced by:  pilem2  22932  pilem3  22933