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Theorem mulcxp 26720

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 1207	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ)
2	1	recnd 11200	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
3	2	mul01d 11372	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐴 · 0) = 0)
4	3	oveq1d 7400	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑_𝑐𝐶) = (0↑_𝑐𝐶))
5		simp3 1147	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
6	2, 5	mulcxplem 26719	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
7	4, 6	eqtrd 2791	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 0)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
8		oveq2 7393	. . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0))
9	8	oveq1d 7400	. . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴 · 0)↑_𝑐𝐶))
10		oveq1 7392	. . . . 5 ⊢ (𝐵 = 0 → (𝐵↑_𝑐𝐶) = (0↑_𝑐𝐶))
11	10	oveq2d 7401	. . . 4 ⊢ (𝐵 = 0 → ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
12	9, 11	eqeq12d 2772	. . 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 1209	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ)
15	14	recnd 11200	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
16	15	mul02d 11371	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0 · 𝐵) = 0)
17	16	oveq1d 7400	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑_𝑐𝐶) = (0↑_𝑐𝐶))
18	15, 5	mulcxplem 26719	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)))
19		cxpcl 26709	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
20	15, 5, 19	syl2anc 592	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
21		0cn 11161	. . . . . . . . 9 ⊢ 0 ∈ ℂ
22		cxpcl 26709	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) ∈ ℂ)
23	21, 5, 22	sylancr 595	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) ∈ ℂ)
24	20, 23	mulcomd 11193	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
25	18, 24	eqtrd 2791	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
26	17, 25	eqtrd 2791	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((0 · 𝐵)↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
27		oveq1 7392	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵))
28	27	oveq1d 7400	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((0 · 𝐵)↑_𝑐𝐶))
29		oveq1 7392	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑_𝑐𝐶) = (0↑_𝑐𝐶))
30	29	oveq1d 7400	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
31	28, 30	eqeq12d 2772	. . . . 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 1235	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐴)
36		simprl 778	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
37	34, 35, 36	ne0gt0d 11310	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐴)
38	34, 37	elrpd 13024	. . . . . . . . . 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 1237	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 ≤ 𝐵)
41		simprr 780	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
42	39, 40, 41	ne0gt0d 11310	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐵)
43	39, 42	elrpd 13024	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℝ⁺)
44	38, 43	relogmuld 26660	. . . . . . . . 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 7401	. . . . . . . 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 26605	. . . . . . . . 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 26605	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐵) ∈ ℂ)
51	46, 48, 50	adddid 11196	. . . . . . . 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 2791	. . . . . . 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 6860	. . . . . 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 11192	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐴)) ∈ ℂ)
55	46, 50	mulcld 11192	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐵)) ∈ ℂ)
56		efadd 16100	. . . . . . 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 592	. . . . . 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 2791	. . . . 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 11192	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ∈ ℂ)
60	47, 49, 36, 41	mulne0d 11829	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0)
61		cxpef 26700	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐵) ≠ 0 ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))))
62	59, 60, 46, 61	syl3anc 1386	. . . . 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 26700	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
64	47, 36, 46, 63	syl3anc 1386	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
65		cxpef 26700	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
66	49, 41, 46, 65	syl3anc 1386	. . . . . 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 7403	. . . . 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 2801	. . . 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 3037	. 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 3037	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 1095 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 class class class wbr 5094 ‘cfv 6510 (class class class)co 7385 ℂcc 11061 ℝcr 11062 0cc0 11063 + caddc 11066 · cmul 11068 ≤ cle 11207 expce 16067 logclog 26589 ↑_𝑐ccxp 26590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-inf2 9586 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-er 8666 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9448 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-dec 12679 df-uz 12830 df-q 12940 df-rp 12984 df-xneg 13104 df-xadd 13105 df-xmul 13106 df-ioo 13343 df-ioc 13344 df-ico 13345 df-icc 13346 df-fz 13503 df-fzo 13650 df-fl 13792 df-mod 13870 df-seq 14005 df-exp 14065 df-fac 14277 df-bc 14306 df-hash 14334 df-shft 15070 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-limsup 15474 df-clim 15491 df-rlim 15492 df-sum 15690 df-ef 16073 df-sin 16075 df-cos 16076 df-pi 16078 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ds 17284 df-unif 17285 df-hom 17286 df-cco 17287 df-rest 17427 df-topn 17428 df-0g 17446 df-gsum 17447 df-topgen 17448 df-pt 17449 df-prds 17452 df-xrs 17508 df-qtop 17513 df-imas 17514 df-xps 17516 df-mre 17590 df-mrc 17591 df-acs 17593 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-submnd 18794 df-mulg 19086 df-cntz 19333 df-cmn 19798 df-psmet 21389 df-xmet 21390 df-met 21391 df-bl 21392 df-mopn 21393 df-fbas 21394 df-fg 21395 df-cnfld 21398 df-top 22927 df-topon 22944 df-topsp 22966 df-bases 22979 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-haus 23348 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24913 df-limc 25901 df-dv 25902 df-log 26591 df-cxp 26592

This theorem is referenced by: cxprec 26721 divcxp 26722 mulcxpd 26763 amgmlemALT 50372

W3C validator