Theorem lcmval

Description: Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N . If either M or N is 0 , the result is defined conventionally as 0 . Contrast with df-gcd and gcdval . (Contributed by Steve Rodriguez, 20-Jan-2020) (Revised by AV, 16-Sep-2020)

	Ref	Expression
Assertion	lcmval	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	|- ( x = M -> ( x = 0 <-> M = 0 ) )
2	1	orbi1d	|- ( x = M -> ( ( x = 0 \/ y = 0 ) <-> ( M = 0 \/ y = 0 ) ) )
3		breq1	|- ( x = M -> ( x || n <-> M || n ) )
4	3	anbi1d	|- ( x = M -> ( ( x || n /\ y || n ) <-> ( M || n /\ y || n ) ) )
5	4	rabbidv	|- ( x = M -> { n e. NN | ( x || n /\ y || n ) } = { n e. NN | ( M || n /\ y || n ) } )
6	5	infeq1d	|- ( x = M -> inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) )
7	2 6	ifbieq2d	|- ( x = M -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) )
8		eqeq1	|- ( y = N -> ( y = 0 <-> N = 0 ) )
9	8	orbi2d	|- ( y = N -> ( ( M = 0 \/ y = 0 ) <-> ( M = 0 \/ N = 0 ) ) )
10		breq1	|- ( y = N -> ( y || n <-> N || n ) )
11	10	anbi2d	|- ( y = N -> ( ( M || n /\ y || n ) <-> ( M || n /\ N || n ) ) )
12	11	rabbidv	|- ( y = N -> { n e. NN | ( M || n /\ y || n ) } = { n e. NN | ( M || n /\ N || n ) } )
13	12	infeq1d	|- ( y = N -> inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
14	9 13	ifbieq2d	|- ( y = N -> if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
15		df-lcm	|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
16		c0ex	|- 0 e. _V
17		ltso	|- < Or RR
18	17	infex	|- inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. _V
19	16 18	ifex	|- if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) e. _V
20	7 14 15 19	ovmpo	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )