Theorem 7prm

Description: 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014) (Revised by Mario Carneiro, 20-Apr-2015)

	Ref	Expression
Assertion	7prm	$$⊢7\in \mathbb{P}$$

Proof

Step	Hyp	Ref	Expression
1		7nn	$$⊢7\in \mathbb{N}$$
2		1lt7	$$⊢1<7$$
3		2nn	$$⊢2\in \mathbb{N}$$
4		3nn0	$$⊢3\in {\mathbb{N}}_0$$
5		1nn	$$⊢1\in \mathbb{N}$$
6		2t3e6	$$⊢2\cdot 3=6$$
7	6	oveq1i	$$⊢2\cdot 3+1=6+1$$
8		df-7	$$⊢7=6+1$$
9	7 8	eqtr4i	$$⊢2\cdot 3+1=7$$
10		1lt2	$$⊢1<2$$
11	3 4 5 9 10	ndvdsi	$$⊢\neg 2\parallel 7$$
12		3nn	$$⊢3\in \mathbb{N}$$
13		2nn0	$$⊢2\in {\mathbb{N}}_0$$
14		3t2e6	$$⊢3\cdot 2=6$$
15	14	oveq1i	$$⊢3\cdot 2+1=6+1$$
16	15 8	eqtr4i	$$⊢3\cdot 2+1=7$$
17		1lt3	$$⊢1<3$$
18	12 13 5 16 17	ndvdsi	$$⊢\neg 3\parallel 7$$
19		5nn0	$$⊢5\in {\mathbb{N}}_0$$
20		7nn0	$$⊢7\in {\mathbb{N}}_0$$
21		7lt10	$$⊢7<10$$
22	3 19 20 21	declti	$$⊢7<25$$
23	1 2 11 18 22	prmlem1	$$⊢7\in \mathbb{P}$$