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Metamath Proof Explorer

Theorem fac3

Description: The factorial of 3. (Contributed by NM, 17-Mar-2005)

		Ref	Expression
	Assertion	fac3	$⊢ 3! = 6$

Proof

Step	Hyp	Ref	Expression
1		df-3	$⊢ 3 = 2 + 1$
2	1	fveq2i	$⊢ 3! = (2 + 1)!$
3		2nn0	$⊢ 2 \in ℕ_{0}$
4		facp1	$⊢ 2 \in ℕ_{0} \to (2 + 1)! = 2! (2 + 1)$
5	3 4	ax-mp	$⊢ (2 + 1)! = 2! (2 + 1)$
6		fac2	$⊢ 2! = 2$
7		2p1e3	$⊢ 2 + 1 = 3$
8	6 7	oveq12i	$⊢ 2! (2 + 1) = 2 \cdot 3$
9		2cn	$⊢ 2 \in ℂ$
10		3cn	$⊢ 3 \in ℂ$
11	9 10	mulcomi	$⊢ 2 \cdot 3 = 3 \cdot 2$
12		3t2e6	$⊢ 3 \cdot 2 = 6$
13	8 11 12	3eqtri	$⊢ 2! (2 + 1) = 6$
14	2 5 13	3eqtri	$⊢ 3! = 6$