Proof of Theorem mulcxp
| Step | Hyp | Ref
| Expression |
| 1 | | simp1l 1198 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ) |
| 2 | 1 | recnd 11289 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) |
| 3 | 2 | mul01d 11460 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 · 0) = 0) |
| 4 | 3 | oveq1d 7446 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑𝑐𝐶) =
(0↑𝑐𝐶)) |
| 5 | | simp3 1139 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) |
| 6 | 2, 5 | mulcxplem 26726 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) →
(0↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
| 7 | 4, 6 | eqtrd 2777 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
| 8 | | oveq2 7439 |
. . . . 5
⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0)) |
| 9 | 8 | oveq1d 7446 |
. . . 4
⊢ (𝐵 = 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴 · 0)↑𝑐𝐶)) |
| 10 | | oveq1 7438 |
. . . . 5
⊢ (𝐵 = 0 → (𝐵↑𝑐𝐶) = (0↑𝑐𝐶)) |
| 11 | 10 | oveq2d 7447 |
. . . 4
⊢ (𝐵 = 0 → ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶))) |
| 12 | 9, 11 | eqeq12d 2753 |
. . 3
⊢ (𝐵 = 0 → (((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)) ↔ ((𝐴 · 0)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (0↑𝑐𝐶)))) |
| 13 | 7, 12 | syl5ibrcom 247 |
. 2
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵 = 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)))) |
| 14 | | simp2l 1200 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ) |
| 15 | 14 | recnd 11289 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) |
| 16 | 15 | mul02d 11459 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0 · 𝐵) = 0) |
| 17 | 16 | oveq1d 7446 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑𝑐𝐶) =
(0↑𝑐𝐶)) |
| 18 | 15, 5 | mulcxplem 26726 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) →
(0↑𝑐𝐶) = ((𝐵↑𝑐𝐶) · (0↑𝑐𝐶))) |
| 19 | | cxpcl 26716 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵↑𝑐𝐶) ∈
ℂ) |
| 20 | 15, 5, 19 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵↑𝑐𝐶) ∈ ℂ) |
| 21 | | 0cn 11253 |
. . . . . . . . 9
⊢ 0 ∈
ℂ |
| 22 | | cxpcl 26716 |
. . . . . . . . 9
⊢ ((0
∈ ℂ ∧ 𝐶
∈ ℂ) → (0↑𝑐𝐶) ∈ ℂ) |
| 23 | 21, 5, 22 | sylancr 587 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) →
(0↑𝑐𝐶) ∈ ℂ) |
| 24 | 20, 23 | mulcomd 11282 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐵↑𝑐𝐶) · (0↑𝑐𝐶)) =
((0↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |
| 25 | 18, 24 | eqtrd 2777 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) →
(0↑𝑐𝐶) = ((0↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |
| 26 | 17, 25 | eqtrd 2777 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑𝑐𝐶) =
((0↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |
| 27 | | oveq1 7438 |
. . . . . . 7
⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵)) |
| 28 | 27 | oveq1d 7446 |
. . . . . 6
⊢ (𝐴 = 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((0 · 𝐵)↑𝑐𝐶)) |
| 29 | | oveq1 7438 |
. . . . . . 7
⊢ (𝐴 = 0 → (𝐴↑𝑐𝐶) = (0↑𝑐𝐶)) |
| 30 | 29 | oveq1d 7446 |
. . . . . 6
⊢ (𝐴 = 0 → ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)) = ((0↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |
| 31 | 28, 30 | eqeq12d 2753 |
. . . . 5
⊢ (𝐴 = 0 → (((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)) ↔ ((0 · 𝐵)↑𝑐𝐶) = ((0↑𝑐𝐶) · (𝐵↑𝑐𝐶)))) |
| 32 | 26, 31 | syl5ibrcom 247 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 = 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)))) |
| 33 | 32 | a1dd 50 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 = 0 → (𝐵 ≠ 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶))))) |
| 34 | 1 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℝ) |
| 35 | | simpl1r 1226 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐴) |
| 36 | | simprl 771 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0) |
| 37 | 34, 35, 36 | ne0gt0d 11398 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐴) |
| 38 | 34, 37 | elrpd 13074 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈
ℝ+) |
| 39 | 14 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℝ) |
| 40 | | simpl2r 1228 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐵) |
| 41 | | simprr 773 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0) |
| 42 | 39, 40, 41 | ne0gt0d 11398 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐵) |
| 43 | 39, 42 | elrpd 13074 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈
ℝ+) |
| 44 | 38, 43 | relogmuld 26667 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵))) |
| 45 | 44 | oveq2d 7447 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘(𝐴 · 𝐵))) = (𝐶 · ((log‘𝐴) + (log‘𝐵)))) |
| 46 | 5 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐶 ∈ ℂ) |
| 47 | 2 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℂ) |
| 48 | 47, 36 | logcld 26612 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐴) ∈
ℂ) |
| 49 | 15 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ) |
| 50 | 49, 41 | logcld 26612 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐵) ∈
ℂ) |
| 51 | 46, 48, 50 | adddid 11285 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · ((log‘𝐴) + (log‘𝐵))) = ((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))) |
| 52 | 45, 51 | eqtrd 2777 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘(𝐴 · 𝐵))) = ((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))) |
| 53 | 52 | fveq2d 6910 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))) = (exp‘((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵))))) |
| 54 | 46, 48 | mulcld 11281 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐴)) ∈ ℂ) |
| 55 | 46, 50 | mulcld 11281 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐵)) ∈ ℂ) |
| 56 | | efadd 16130 |
. . . . . . 7
⊢ (((𝐶 · (log‘𝐴)) ∈ ℂ ∧ (𝐶 · (log‘𝐵)) ∈ ℂ) →
(exp‘((𝐶 ·
(log‘𝐴)) + (𝐶 · (log‘𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵))))) |
| 57 | 54, 55, 56 | syl2anc 584 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (exp‘((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵))))) |
| 58 | 53, 57 | eqtrd 2777 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵))))) |
| 59 | 47, 49 | mulcld 11281 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ∈ ℂ) |
| 60 | 47, 49, 36, 41 | mulne0d 11915 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0) |
| 61 | | cxpef 26707 |
. . . . . 6
⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐵) ≠ 0 ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵))))) |
| 62 | 59, 60, 46, 61 | syl3anc 1373 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 · 𝐵)↑𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵))))) |
| 63 | | cxpef 26707 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴)))) |
| 64 | 47, 36, 46, 63 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴)))) |
| 65 | | cxpef 26707 |
. . . . . . 7
⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵)))) |
| 66 | 49, 41, 46, 65 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐵↑𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵)))) |
| 67 | 64, 66 | oveq12d 7449 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵))))) |
| 68 | 58, 62, 67 | 3eqtr4d 2787 |
. . . 4
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |
| 69 | 68 | exp32 420 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 ≠ 0 → (𝐵 ≠ 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶))))) |
| 70 | 33, 69 | pm2.61dne 3028 |
. 2
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵 ≠ 0 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)))) |
| 71 | 13, 70 | pm2.61dne 3028 |
1
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶))) |