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Metamath Proof Explorer

Theorem cos2pi

Description: The cosine of 2 _pi is 1. (Contributed by Paul Chapman, 23-Jan-2008)

Ref Expression
Assertion cos2pi ( cos ‘ ( 2 · π ) ) = 1

Proof

Step Hyp Ref Expression
1 picn ⊢ π ∈ ℂ
2 cos2t ⊢ ( π ∈ ℂ → ( cos ‘ ( 2 · π ) ) = ( ( 2 · ( ( cos ‘ π ) ↑ 2 ) ) − 1 ) )
3 1 2 ax-mp ⊢ ( cos ‘ ( 2 · π ) ) = ( ( 2 · ( ( cos ‘ π ) ↑ 2 ) ) − 1 )
4 cospi ⊢ ( cos ‘ π ) = - 1
5 4 oveq1i ⊢ ( ( cos ‘ π ) ↑ 2 ) = ( - 1 ↑ 2 )
6 ax-1cn ⊢ 1 ∈ ℂ
7 sqneg ⊢ ( 1 ∈ ℂ → ( - 1 ↑ 2 ) = ( 1 ↑ 2 ) )
8 6 7 ax-mp ⊢ ( - 1 ↑ 2 ) = ( 1 ↑ 2 )
9 sq1 ⊢ ( 1 ↑ 2 ) = 1
10 5 8 9 3eqtri ⊢ ( ( cos ‘ π ) ↑ 2 ) = 1
11 10 oveq2i ⊢ ( 2 · ( ( cos ‘ π ) ↑ 2 ) ) = ( 2 · 1 )
12 2t1e2 ⊢ ( 2 · 1 ) = 2
13 11 12 eqtri ⊢ ( 2 · ( ( cos ‘ π ) ↑ 2 ) ) = 2
14 13 oveq1i ⊢ ( ( 2 · ( ( cos ‘ π ) ↑ 2 ) ) − 1 ) = ( 2 − 1 )
15 2m1e1 ⊢ ( 2 − 1 ) = 1
16 3 14 15 3eqtri ⊢ ( cos ‘ ( 2 · π ) ) = 1