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 ∈ ℙ

Proof

Step	Hyp	Ref	Expression
1		7nn	⊢ 7 ∈ ℕ
2		1lt7	⊢ 1 < 7
3		2nn	⊢ 2 ∈ ℕ
4		3nn0	⊢ 3 ∈ ℕ0
5		1nn	⊢ 1 ∈ ℕ
6		2t3e6	⊢ ( 2 · 3 ) = 6
7	6	oveq1i	⊢ ( ( 2 · 3 ) + 1 ) = ( 6 + 1 )
8		df-7	⊢ 7 = ( 6 + 1 )
9	7 8	eqtr4i	⊢ ( ( 2 · 3 ) + 1 ) = 7
10		1lt2	⊢ 1 < 2
11	3 4 5 9 10	ndvdsi	⊢ ¬ 2 ∥ 7
12		3nn	⊢ 3 ∈ ℕ
13		2nn0	⊢ 2 ∈ ℕ0
14		3t2e6	⊢ ( 3 · 2 ) = 6
15	14	oveq1i	⊢ ( ( 3 · 2 ) + 1 ) = ( 6 + 1 )
16	15 8	eqtr4i	⊢ ( ( 3 · 2 ) + 1 ) = 7
17		1lt3	⊢ 1 < 3
18	12 13 5 16 17	ndvdsi	⊢ ¬ 3 ∥ 7
19		5nn0	⊢ 5 ∈ ℕ0
20		7nn0	⊢ 7 ∈ ℕ0
21		7lt10	⊢ 7 < ; 1 0
22	3 19 20 21	declti	⊢ 7 < ; 2 5
23	1 2 11 18 22	prmlem1	⊢ 7 ∈ ℙ