Metamath Proof Explorer

Theorem abscxp2

Description: Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion abscxp2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) )

Proof

Step Hyp Ref Expression
1 0red ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → 0 ∈ ℝ )
2 0le0 ⊢ 0 ≤ 0
3 2 a1i ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → 0 ≤ 0 )
4 simplr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → 𝐵 ∈ ℝ )
5 recxpcl ⊢ ( ( 0 ∈ ℝ ∧ 0 ≤ 0 ∧ 𝐵 ∈ ℝ ) → ( 0 ↑𝑐 𝐵 ) ∈ ℝ )
6 1 3 4 5 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( 0 ↑𝑐 𝐵 ) ∈ ℝ )
7 cxpge0 ⊢ ( ( 0 ∈ ℝ ∧ 0 ≤ 0 ∧ 𝐵 ∈ ℝ ) → 0 ≤ ( 0 ↑𝑐 𝐵 ) )
8 1 3 4 7 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → 0 ≤ ( 0 ↑𝑐 𝐵 ) )
9 6 8 absidd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( abs ‘ ( 0 ↑𝑐 𝐵 ) ) = ( 0 ↑𝑐 𝐵 ) )
10 simpr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → 𝐴 = 0 )
11 10 oveq1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( 𝐴 ↑𝑐 𝐵 ) = ( 0 ↑𝑐 𝐵 ) )
12 11 fveq2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( abs ‘ ( 0 ↑𝑐 𝐵 ) ) )
13 10 abs00bd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( abs ‘ 𝐴 ) = 0 )
14 13 oveq1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) = ( 0 ↑𝑐 𝐵 ) )
15 9 12 14 3eqtr4d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 = 0 ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) )
16 simplr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → 𝐵 ∈ ℝ )
17 16 recnd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → 𝐵 ∈ ℂ )
18 logcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )
19 18 adantlr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )
20 17 19 mulcld ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ )
21 absef ⊢ ( ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ → ( abs ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( ℜ ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
22 20 21 syl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( ℜ ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
23 16 19 remul2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) = ( 𝐵 · ( ℜ ‘ ( log ‘ 𝐴 ) ) ) )
24 relog ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( log ‘ 𝐴 ) ) = ( log ‘ ( abs ‘ 𝐴 ) ) )
25 24 adantlr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( log ‘ 𝐴 ) ) = ( log ‘ ( abs ‘ 𝐴 ) ) )
26 25 oveq2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( 𝐵 · ( ℜ ‘ ( log ‘ 𝐴 ) ) ) = ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) )
27 23 26 eqtrd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) = ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) )
28 27 fveq2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( ℜ ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) ) )
29 22 28 eqtrd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) ) )
30 simpll ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℂ )
31 simpr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → 𝐴 ≠ 0 )
32 cxpef ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
33 30 31 17 32 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
34 33 fveq2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( abs ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
35 30 abscld ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ )
36 35 recnd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℂ )
37 abs00 ⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
38 37 adantr ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
39 38 necon3bid ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )
40 39 biimpar ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ≠ 0 )
41 cxpef ⊢ ( ( ( abs ‘ 𝐴 ) ∈ ℂ ∧ ( abs ‘ 𝐴 ) ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) ) )
42 36 40 17 41 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ ( abs ‘ 𝐴 ) ) ) ) )
43 29 34 42 3eqtr4d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) )
44 15 43 pm2.61dane ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( ( abs ‘ 𝐴 ) ↑𝑐 𝐵 ) )