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Theorem mulcxp 25270

Description: Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
mulcxp	⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))

**Proof of Theorem mulcxp**
Step	Hyp	Ref	Expression
1		simp1l 1193	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ)
2	1	recnd 10671	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
3	2	mul01d 10841	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 · 0) = 0)
4	3	oveq1d 7173	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑_𝑐𝐶) = (0↑_𝑐𝐶))
5		simp3 1134	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
6	2, 5	mulcxplem 25269	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
7	4, 6	eqtrd 2858	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
8		oveq2 7166	. . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0))
9	8	oveq1d 7173	. . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴 · 0)↑_𝑐𝐶))
10		oveq1 7165	. . . . 5 ⊢ (𝐵 = 0 → (𝐵↑_𝑐𝐶) = (0↑_𝑐𝐶))
11	10	oveq2d 7174	. . . 4 ⊢ (𝐵 = 0 → ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
12	9, 11	eqeq12d 2839	. . 3 ⊢ (𝐵 = 0 → (((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) ↔ ((𝐴 · 0)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶))))
13	7, 12	syl5ibrcom 249	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵 = 0 → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶))))
14		simp2l 1195	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ)
15	14	recnd 10671	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
16	15	mul02d 10840	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0 · 𝐵) = 0)
17	16	oveq1d 7173	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑_𝑐𝐶) = (0↑_𝑐𝐶))
18	15, 5	mulcxplem 25269	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)))
19		cxpcl 25259	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
20	15, 5, 19	syl2anc 586	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
21		0cn 10635	. . . . . . . . 9 ⊢ 0 ∈ ℂ
22		cxpcl 25259	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) ∈ ℂ)
23	21, 5, 22	sylancr 589	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) ∈ ℂ)
24	20, 23	mulcomd 10664	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
25	18, 24	eqtrd 2858	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
26	17, 25	eqtrd 2858	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
27		oveq1 7165	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵))
28	27	oveq1d 7173	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((0 · 𝐵)↑_𝑐𝐶))
29		oveq1 7165	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑_𝑐𝐶) = (0↑_𝑐𝐶))
30	29	oveq1d 7173	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
31	28, 30	eqeq12d 2839	. . . . 5 ⊢ (𝐴 = 0 → (((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) ↔ ((0 · 𝐵)↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶))))
32	26, 31	syl5ibrcom 249	. . . 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 483	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℝ)
35		simpl1r 1221	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐴)
36		simprl 769	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
37	34, 35, 36	ne0gt0d 10779	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐴)
38	34, 37	elrpd 12431	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℝ⁺)
39	14	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℝ)
40		simpl2r 1223	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐵)
41		simprr 771	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
42	39, 40, 41	ne0gt0d 10779	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐵)
43	39, 42	elrpd 12431	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℝ⁺)
44	38, 43	relogmuld 25210	. . . . . . . . 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 7174	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐶 ∈ ℂ)
47	2	adantr 483	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℂ)
48	47, 36	logcld 25156	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐴) ∈ ℂ)
49	15	adantr 483	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ)
50	49, 41	logcld 25156	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐵) ∈ ℂ)
51	46, 48, 50	adddid 10667	. . . . . . . 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 2858	. . . . . . 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 6676	. . . . . 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 10663	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐴)) ∈ ℂ)
55	46, 50	mulcld 10663	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐵)) ∈ ℂ)
56		efadd 15449	. . . . . . 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 586	. . . . . 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 2858	. . . . 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 10663	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ∈ ℂ)
60	47, 49, 36, 41	mulne0d 11294	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0)
61		cxpef 25250	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐵) ≠ 0 ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))))
62	59, 60, 46, 61	syl3anc 1367	. . . . 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 25250	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
64	47, 36, 46, 63	syl3anc 1367	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
65		cxpef 25250	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
66	49, 41, 46, 65	syl3anc 1367	. . . . . 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 7176	. . . . 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 2868	. . . 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 423	. . 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 3105	. 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 3105	1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 ≤ cle 10678 expce 15417 logclog 25140 ↑_𝑐ccxp 25141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-cxp 25143

This theorem is referenced by: cxprec 25271 divcxp 25272 mulcxpd 25313 amgmlemALT 44911

W3C validator