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Theorem mulcxp 26742

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 1196	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ)
2	1	recnd 11287	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
3	2	mul01d 11458	. . . . 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 1137	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
6	2, 5	mulcxplem 26741	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (0↑_𝑐𝐶)))
7	4, 6	eqtrd 2775	. . 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 2751	. . 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 1198	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ)
15	14	recnd 11287	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
16	15	mul02d 11457	. . . . . . 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 26741	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)))
19		cxpcl 26731	. . . . . . . . 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 11251	. . . . . . . . 9 ⊢ 0 ∈ ℂ
22		cxpcl 26731	. . . . . . . . 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 11280	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐵↑_𝑐𝐶) · (0↑_𝑐𝐶)) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
25	18, 24	eqtrd 2775	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → (0↑_𝑐𝐶) = ((0↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))
26	17, 25	eqtrd 2775	. . . . 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 2751	. . . . 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 1224	. . . . . . . . . . . 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 11396	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐴)
38	34, 37	elrpd 13072	. . . . . . . . . 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 1226	. . . . . . . . . . . 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 11396	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 0 < 𝐵)
43	39, 42	elrpd 13072	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℝ⁺)
44	38, 43	relogmuld 26682	. . . . . . . . 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 26627	. . . . . . . . 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 26627	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (log‘𝐵) ∈ ℂ)
51	46, 48, 50	adddid 11283	. . . . . . . 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 2775	. . . . . . 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 6911	. . . . . 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 11279	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐴)) ∈ ℂ)
55	46, 50	mulcld 11279	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐶 · (log‘𝐵)) ∈ ℂ)
56		efadd 16127	. . . . . . 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 2775	. . . . 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 11279	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ∈ ℂ)
60	47, 49, 36, 41	mulne0d 11913	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0)
61		cxpef 26722	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ (𝐴 · 𝐵) ≠ 0 ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))))
62	59, 60, 46, 61	syl3anc 1370	. . . . 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 26722	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
64	47, 36, 46, 63	syl3anc 1370	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
65		cxpef 26722	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
66	49, 41, 46, 65	syl3anc 1370	. . . . . 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 2785	. . . 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 3026	. 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 3026	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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 · cmul 11158 ≤ cle 11294 expce 16094 logclog 26611 ↑_𝑐ccxp 26612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614

This theorem is referenced by: cxprec 26743 divcxp 26744 mulcxpd 26785 amgmlemALT 49034

W3C validator