Theorem lcm1un

Description: Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024)

	Ref	Expression
Assertion	lcm1un	|- ( _lcm ` ( 1 ... 1 ) ) = 1

Proof

Step	Hyp	Ref	Expression
1		1nn	|- 1 e. NN
2		id	|- ( 1 e. NN -> 1 e. NN )
3	2	lcmfunnnd	|- ( 1 e. NN -> ( _lcm ` ( 1 ... 1 ) ) = ( ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) lcm 1 ) )
4	1 3	ax-mp	|- ( _lcm ` ( 1 ... 1 ) ) = ( ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) lcm 1 )
5		1m1e0	|- ( 1 - 1 ) = 0
6	5	oveq2i	|- ( 1 ... ( 1 - 1 ) ) = ( 1 ... 0 )
7		fz10	|- ( 1 ... 0 ) = (/)
8	6 7	eqtri	|- ( 1 ... ( 1 - 1 ) ) = (/)
9	8	fveq2i	|- ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) = ( _lcm ` (/) )
10		lcmf0	|- ( _lcm ` (/) ) = 1
11	9 10	eqtri	|- ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) = 1
12	11	oveq1i	|- ( ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) lcm 1 ) = ( 1 lcm 1 )
13		1z	|- 1 e. ZZ
14		lcmid	|- ( 1 e. ZZ -> ( 1 lcm 1 ) = ( abs ` 1 ) )
15	13 14	ax-mp	|- ( 1 lcm 1 ) = ( abs ` 1 )
16		abs1	|- ( abs ` 1 ) = 1
17	15 16	eqtri	|- ( 1 lcm 1 ) = 1
18	12 17	eqtri	|- ( ( _lcm ` ( 1 ... ( 1 - 1 ) ) ) lcm 1 ) = 1
19	4 18	eqtri	|- ( _lcm ` ( 1 ... 1 ) ) = 1