Definition df-gcd 15217

Description: Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 27314). For an alternate definition, based on the definition in [ApostolNT] p. 15, see dfgcd2 15263. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
df-gcd	|- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )

Distinct variable group:   x, n, y

Detailed syntax breakdown of Definition df-gcd

Step	Hyp	Ref	Expression
1		cgcd 15216	. 2 class gcd
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cz 11377	. . 3 class ZZ
5	2	cv 1482	. . . . . 6 class x
6		cc0 9936	. . . . . 6 class 0
7	5, 6	wceq 1483	. . . . 5 wff x = 0
8	3	cv 1482	. . . . . 6 class y
9	8, 6	wceq 1483	. . . . 5 wff y = 0
10	7, 9	wa 384	. . . 4 wff ( x = 0 /\ y = 0 )
11		vn	. . . . . . . . 9 setvar n
12	11	cv 1482	. . . . . . . 8 class n
13		cdvds 14983	. . . . . . . 8 class ||
14	12, 5, 13	wbr 4653	. . . . . . 7 wff n || x
15	12, 8, 13	wbr 4653	. . . . . . 7 wff n || y
16	14, 15	wa 384	. . . . . 6 wff ( n || x /\ n || y )
17	16, 11, 4	crab 2916	. . . . 5 class { n e. ZZ | ( n || x /\ n || y ) }
18		cr 9935	. . . . 5 class RR
19		clt 10074	. . . . 5 class <
20	17, 18, 19	csup 8346	. . . 4 class sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < )
21	10, 6, 20	cif 4086	. . 3 class if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) )
22	2, 3, 4, 4, 21	cmpt2 6652	. 2 class ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
23	1, 22	wceq 1483	1 wff gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )

This definition is referenced by:  gcdval  15218  gcdf  15234