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Metamath Proof Explorer

Theorem i4

Description: _i to the fourth power. (Contributed by NM, 31-Jan-2007)

Ref Expression
Assertion i4 ( i ↑ 4 ) = 1

Proof

Step Hyp Ref Expression
1 ax-icn ⊢ i ∈ ℂ
2 2nn0 ⊢ 2 ∈ ℕ0
3 expadd ⊢ ( ( i ∈ ℂ ∧ 2 ∈ ℕ0 ∧ 2 ∈ ℕ0 ) → ( i ↑ ( 2 + 2 ) ) = ( ( i ↑ 2 ) · ( i ↑ 2 ) ) )
4 1 2 2 3 mp3an ⊢ ( i ↑ ( 2 + 2 ) ) = ( ( i ↑ 2 ) · ( i ↑ 2 ) )
5 2p2e4 ⊢ ( 2 + 2 ) = 4
6 5 oveq2i ⊢ ( i ↑ ( 2 + 2 ) ) = ( i ↑ 4 )
7 i2 ⊢ ( i ↑ 2 ) = - 1
8 7 7 oveq12i ⊢ ( ( i ↑ 2 ) · ( i ↑ 2 ) ) = ( - 1 · - 1 )
9 ax-1cn ⊢ 1 ∈ ℂ
10 9 9 mul2negi ⊢ ( - 1 · - 1 ) = ( 1 · 1 )
11 1t1e1 ⊢ ( 1 · 1 ) = 1
12 8 10 11 3eqtri ⊢ ( ( i ↑ 2 ) · ( i ↑ 2 ) ) = 1
13 4 6 12 3eqtr3i ⊢ ( i ↑ 4 ) = 1