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Metamath Proof Explorer

Theorem cxpp1

Description: Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxpp1 ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ax-1cn ⊢ 1 ∈ ℂ
2 cxpadd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
3 1 2 mp3an3 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
4 3 3impa ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
5 cxp1 ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑𝑐 1 ) = 𝐴 )
6 5 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 1 ) = 𝐴 )
7 6 oveq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )
8 4 7 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )