Theorem cxpneg

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

	Ref	Expression
Assertion	cxpneg	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ )
2		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
3		cxpcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )
4	1 2 3	syl2anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )
5	2	negcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → - 𝐵 ∈ ℂ )
6		cxpcl	⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) ∈ ℂ )
7	1 5 6	syl2anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) ∈ ℂ )
8		cxpne0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 )
9	2	negidd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + - 𝐵 ) = 0 )
10	9	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( 𝐴 ↑𝑐 0 ) )
11		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ≠ 0 )
12		cxpadd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) )
13	1 11 2 5 12	syl211anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) )
14		cxp0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑𝑐 0 ) = 1 )
15	1 14	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 0 ) = 1 )
16	10 13 15	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) = 1 )
17	4 7 8 16	mvllmuld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )