Definition df-bits 15144

Description: Define the binary bits of an integer. The expression M e. (bits ` N ) means that the M-th bit of N is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
df-bits	|- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )

Distinct variable group:   m, n

Detailed syntax breakdown of Definition df-bits

Step	Hyp	Ref	Expression
1		cbits 15141	. 2 class bits
2		vn	. . 3 setvar n
3		cz 11377	. . 3 class ZZ
4		c2 11070	. . . . . 6 class 2
5	2	cv 1482	. . . . . . . 8 class n
6		vm	. . . . . . . . . 10 setvar m
7	6	cv 1482	. . . . . . . . 9 class m
8		cexp 12860	. . . . . . . . 9 class ^
9	4, 7, 8	co 6650	. . . . . . . 8 class ( 2 ^ m )
10		cdiv 10684	. . . . . . . 8 class /
11	5, 9, 10	co 6650	. . . . . . 7 class ( n / ( 2 ^ m ) )
12		cfl 12591	. . . . . . 7 class |_
13	11, 12	cfv 5888	. . . . . 6 class ( |_ ` ( n / ( 2 ^ m ) ) )
14		cdvds 14983	. . . . . 6 class ||
15	4, 13, 14	wbr 4653	. . . . 5 wff 2 || ( |_ ` ( n / ( 2 ^ m ) ) )
16	15	wn 3	. . . 4 wff -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) )
17		cn0 11292	. . . 4 class NN0
18	16, 6, 17	crab 2916	. . 3 class { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) }
19	2, 3, 18	cmpt 4729	. 2 class ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
20	1, 19	wceq 1483	1 wff bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )

This definition is referenced by:  bitsfval  15145  bitsval  15146  bitsf  15149