Definition df-lcm 12632

Description: Define the lcm operator. For example, ( 6 lcm 9 ) = 1 8. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)

Assertion

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

Distinct variable group:   x, n, y

Detailed syntax breakdown of Definition df-lcm

Step	Hyp	Ref	Expression
1		clcm 12631	. 2 class lcm
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cz 9478	. . 3 class ZZ
5	2	cv 1396	. . . . . 6 class x
6		cc0 8031	. . . . . 6 class 0
7	5, 6	wceq 1397	. . . . 5 wff x = 0
8	3	cv 1396	. . . . . 6 class y
9	8, 6	wceq 1397	. . . . 5 wff y = 0
10	7, 9	wo 715	. . . 4 wff ( x = 0 \/ y = 0 )
11		vn	. . . . . . . . 9 setvar n
12	11	cv 1396	. . . . . . . 8 class n
13		cdvds 12347	. . . . . . . 8 class ||
14	5, 12, 13	wbr 4088	. . . . . . 7 wff x || n
15	8, 12, 13	wbr 4088	. . . . . . 7 wff y || n
16	14, 15	wa 104	. . . . . 6 wff ( x || n /\ y || n )
17		cn 9142	. . . . . 6 class NN
18	16, 11, 17	crab 2514	. . . . 5 class { n e. NN | ( x || n /\ y || n ) }
19		cr 8030	. . . . 5 class RR
20		clt 8213	. . . . 5 class <
21	18, 19, 20	cinf 7181	. . . 4 class inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < )
22	10, 6, 21	cif 3605	. . 3 class if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) )
23	2, 3, 4, 4, 22	cmpo 6019	. 2 class ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
24	1, 23	wceq 1397	1 wff lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )

This definition is referenced by:  lcmval  12634