Theorem lcm6un

Description: Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024)

	Ref	Expression
Assertion	lcm6un	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6\right)\right)=60$$

Proof

Step	Hyp	Ref	Expression
1		6nn	$$⊢6\in \mathbb{N}$$
2	1	a1i	$$⊢6\in \mathbb{N}\to 6\in \mathbb{N}$$
3	2	lcmfunnnd	$$⊢6\in \mathbb{N}\to \underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6\right)\right)=\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6-1\right)\right)\mathrm{lcm}6$$
4	1 3	ax-mp	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6\right)\right)=\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6-1\right)\right)\mathrm{lcm}6$$
5		6m1e5	$$⊢6-1=5$$
6	5	oveq2i	$$⊢\left(1\dots 6-1\right)=\left(1\dots 5\right)$$
7	6	fveq2i	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6-1\right)\right)=\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 5\right)\right)$$
8	7	oveq1i	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6-1\right)\right)\mathrm{lcm}6=\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 5\right)\right)\mathrm{lcm}6$$
9		lcm5un	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 5\right)\right)=60$$
10	9	oveq1i	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 5\right)\right)\mathrm{lcm}6=60\mathrm{lcm}6$$
11	8 10	eqtri	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6-1\right)\right)\mathrm{lcm}6=60\mathrm{lcm}6$$
12		60lcm6e60	$$⊢60\mathrm{lcm}6=60$$
13	4 11 12	3eqtri	$$⊢\underset{\_}{\mathrm{lcm}}\left(\left(1\dots 6\right)\right)=60$$