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Metamath Proof Explorer

Theorem fac2

Description: The factorial of 2. (Contributed by NM, 17-Mar-2005)

		Ref	Expression
	Assertion	fac2	$⊢ 2! = 2$

Proof

Step	Hyp	Ref	Expression
1		df-2	$⊢ 2 = 1 + 1$
2	1	fveq2i	$⊢ 2! = (1 + 1)!$
3		1nn0	$⊢ 1 \in ℕ_{0}$
4		facp1	$⊢ 1 \in ℕ_{0} \to (1 + 1)! = 1! (1 + 1)$
5	3 4	ax-mp	$⊢ (1 + 1)! = 1! (1 + 1)$
6		fac1	$⊢ 1! = 1$
7		1p1e2	$⊢ 1 + 1 = 2$
8	6 7	oveq12i	$⊢ 1! (1 + 1) = 1 \cdot 2$
9		2cn	$⊢ 2 \in ℂ$
10	9	mullidi	$⊢ 1 \cdot 2 = 2$
11	8 10	eqtri	$⊢ 1! (1 + 1) = 2$
12	5 11	eqtri	$⊢ (1 + 1)! = 2$
13	2 12	eqtri	$⊢ 2! = 2$