Step |
Hyp |
Ref |
Expression |
1 |
|
recn |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ ) |
2 |
|
recn |
⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ ) |
3 |
|
mulcom |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
4 |
1 2 3
|
syl2an |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
5 |
4
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
6 |
5
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( 𝐴 ↑𝑐 ( 𝐶 · 𝐵 ) ) ) |
7 |
|
cxpmul |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) ) |
8 |
2 7
|
syl3an3 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) ) |
9 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ+ ) |
10 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ ) |
11 |
1
|
3ad2ant2 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℂ ) |
12 |
9 10 11
|
cxpmuld |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 ( 𝐶 · 𝐵 ) ) = ( ( 𝐴 ↑𝑐 𝐶 ) ↑𝑐 𝐵 ) ) |
13 |
6 8 12
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) ↑𝑐 𝐵 ) ) |