Theorem absef

Description: The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007)

	Ref	Expression
Assertion	absef	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ 𝐴 ) ) = ( exp ‘ ( ℜ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		replim	⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) )
2	1	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) = ( exp ‘ ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) )
3		recl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
4	3	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
5		ax-icn	⊢ i ∈ ℂ
6		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
7	6	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
8		mulcl	⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
9	5 7 8	sylancr	⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
10		efadd	⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) → ( exp ‘ ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( exp ‘ ( ℜ ‘ 𝐴 ) ) · ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) )
11	4 9 10	syl2anc	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( exp ‘ ( ℜ ‘ 𝐴 ) ) · ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) )
12	2 11	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) = ( ( exp ‘ ( ℜ ‘ 𝐴 ) ) · ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) )
13	12	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ 𝐴 ) ) = ( abs ‘ ( ( exp ‘ ( ℜ ‘ 𝐴 ) ) · ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) )
14	3	reefcld	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ℜ ‘ 𝐴 ) ) ∈ ℝ )
15	14	recnd	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ℜ ‘ 𝐴 ) ) ∈ ℂ )
16		efcl	⊢ ( ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ → ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ∈ ℂ )
17	9 16	syl	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ∈ ℂ )
18	15 17	absmuld	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ( exp ‘ ( ℜ ‘ 𝐴 ) ) · ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) = ( ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) · ( abs ‘ ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) )
19		absefi	⊢ ( ( ℑ ‘ 𝐴 ) ∈ ℝ → ( abs ‘ ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) = 1 )
20	6 19	syl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) = 1 )
21	20	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) · ( abs ‘ ( exp ‘ ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) = ( ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) · 1 ) )
22	13 18 21	3eqtrd	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ 𝐴 ) ) = ( ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) · 1 ) )
23	15	abscld	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) ∈ ℝ )
24	23	recnd	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) ∈ ℂ )
25	24	mulridd	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) · 1 ) = ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) )
26		efgt0	⊢ ( ( ℜ ‘ 𝐴 ) ∈ ℝ → 0 < ( exp ‘ ( ℜ ‘ 𝐴 ) ) )
27	3 26	syl	⊢ ( 𝐴 ∈ ℂ → 0 < ( exp ‘ ( ℜ ‘ 𝐴 ) ) )
28		0re	⊢ 0 ∈ ℝ
29		ltle	⊢ ( ( 0 ∈ ℝ ∧ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) → ( 0 < ( exp ‘ ( ℜ ‘ 𝐴 ) ) → 0 ≤ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) )
30	28 14 29	sylancr	⊢ ( 𝐴 ∈ ℂ → ( 0 < ( exp ‘ ( ℜ ‘ 𝐴 ) ) → 0 ≤ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) )
31	27 30	mpd	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( exp ‘ ( ℜ ‘ 𝐴 ) ) )
32	14 31	absidd	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ ( ℜ ‘ 𝐴 ) ) ) = ( exp ‘ ( ℜ ‘ 𝐴 ) ) )
33	22 25 32	3eqtrd	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( exp ‘ 𝐴 ) ) = ( exp ‘ ( ℜ ‘ 𝐴 ) ) )