cxpsqrt - Metamath Proof Explorer

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Theorem cxpsqrt 26211

Description: The complex exponential function with exponent 1 / 2 exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
cxpsqrt	⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))

**Proof of Theorem cxpsqrt**
Step	Hyp	Ref	Expression
1		halfcn 12427	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
2		halfre 12426	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
3		halfgt0 12428	. . . . . . 7 ⊢ 0 < (1 / 2)
4	2, 3	gt0ne0ii 11750	. . . . . 6 ⊢ (1 / 2) ≠ 0
5		0cxp 26174	. . . . . 6 ⊢ (((1 / 2) ∈ ℂ ∧ (1 / 2) ≠ 0) → (0↑_𝑐(1 / 2)) = 0)
6	1, 4, 5	mp2an 691	. . . . 5 ⊢ (0↑_𝑐(1 / 2)) = 0
7		sqrt0 15188	. . . . 5 ⊢ (√‘0) = 0
8	6, 7	eqtr4i 2764	. . . 4 ⊢ (0↑_𝑐(1 / 2)) = (√‘0)
9		oveq1 7416	. . . 4 ⊢ (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (0↑_𝑐(1 / 2)))
10		fveq2 6892	. . . 4 ⊢ (𝐴 = 0 → (√‘𝐴) = (√‘0))
11	8, 9, 10	3eqtr4a 2799	. . 3 ⊢ (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
12	11	a1i 11	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
13		ax-icn 11169	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
14		sqrtcl 15308	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
15	14	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
16		sqmul 14084	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
17	13, 15, 16	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
18		i2 14166	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
19	18	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i↑2) = -1)
20		sqrtth 15311	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
21	20	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘𝐴)↑2) = 𝐴)
22	19, 21	oveq12d 7427	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i↑2) · ((√‘𝐴)↑2)) = (-1 · 𝐴))
23		mulm1 11655	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴)
24	23	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-1 · 𝐴) = -𝐴)
25	17, 22, 24	3eqtrd 2777	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = -𝐴)
26		cxpsqrtlem 26210	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)
27	26	resqcld 14090	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ∈ ℝ)
28	25, 27	eqeltrrd 2835	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ)
29		negeq0 11514	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))
30	29	necon3bid 2986	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))
31	30	biimpa 478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -𝐴 ≠ 0)
32	31	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ≠ 0)
33	25, 32	eqnetrd 3009	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ≠ 0)
34		sq0i 14157	. . . . . . . . . . . . . . . . . 18 ⊢ ((i · (√‘𝐴)) = 0 → ((i · (√‘𝐴))↑2) = 0)
35	34	necon3i 2974	. . . . . . . . . . . . . . . . 17 ⊢ (((i · (√‘𝐴))↑2) ≠ 0 → (i · (√‘𝐴)) ≠ 0)
36	33, 35	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ≠ 0)
37	26, 36	sqgt0d 14213	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < ((i · (√‘𝐴))↑2))
38	37, 25	breqtrd 5175	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < -𝐴)
39	28, 38	elrpd 13013	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ⁺)
40		logneg 26096	. . . . . . . . . . . . 13 ⊢ (-𝐴 ∈ ℝ⁺ → (log‘--𝐴) = ((log‘-𝐴) + (i · π)))
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘--𝐴) = ((log‘-𝐴) + (i · π)))
42		negneg 11510	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
43	42	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → --𝐴 = 𝐴)
44	43	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘--𝐴) = (log‘𝐴))
45	39	relogcld 26131	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘-𝐴) ∈ ℝ)
46	45	recnd 11242	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘-𝐴) ∈ ℂ)
47		picn 25969	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
48	13, 47	mulcli 11221	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
49		addcom 11400	. . . . . . . . . . . . 13 ⊢ (((log‘-𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘-𝐴) + (i · π)) = ((i · π) + (log‘-𝐴)))
50	46, 48, 49	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((log‘-𝐴) + (i · π)) = ((i · π) + (log‘-𝐴)))
51	41, 44, 50	3eqtr3d 2781	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘𝐴) = ((i · π) + (log‘-𝐴)))
52	51	oveq2d 7425	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((1 / 2) · ((i · π) + (log‘-𝐴))))
53		adddi 11199	. . . . . . . . . . 11 ⊢ (((1 / 2) ∈ ℂ ∧ (i · π) ∈ ℂ ∧ (log‘-𝐴) ∈ ℂ) → ((1 / 2) · ((i · π) + (log‘-𝐴))) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
54	1, 48, 46, 53	mp3an12i 1466	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · ((i · π) + (log‘-𝐴))) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
55	52, 54	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
56		2cn 12287	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
57		2ne0 12316	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
58		divrec2 11889	. . . . . . . . . . . 12 ⊢ (((i · π) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((i · π) / 2) = ((1 / 2) · (i · π)))
59	48, 56, 57, 58	mp3an 1462	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = ((1 / 2) · (i · π))
60	13, 47, 56, 57	divassi 11970	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = (i · (π / 2))
61	59, 60	eqtr3i 2763	. . . . . . . . . 10 ⊢ ((1 / 2) · (i · π)) = (i · (π / 2))
62	61	oveq1i 7419	. . . . . . . . 9 ⊢ (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))) = ((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))
63	55, 62	eqtrdi 2789	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((i · (π / 2)) + ((1 / 2) · (log‘-𝐴))))
64	63	fveq2d 6896	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (exp‘((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))))
65	47, 56, 57	divcli 11956	. . . . . . . . 9 ⊢ (π / 2) ∈ ℂ
66	13, 65	mulcli 11221	. . . . . . . 8 ⊢ (i · (π / 2)) ∈ ℂ
67		mulcl 11194	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘-𝐴) ∈ ℂ) → ((1 / 2) · (log‘-𝐴)) ∈ ℂ)
68	1, 46, 67	sylancr 588	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘-𝐴)) ∈ ℂ)
69		efadd 16037	. . . . . . . 8 ⊢ (((i · (π / 2)) ∈ ℂ ∧ ((1 / 2) · (log‘-𝐴)) ∈ ℂ) → (exp‘((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))) = ((exp‘(i · (π / 2))) · (exp‘((1 / 2) · (log‘-𝐴)))))
70	66, 68, 69	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))) = ((exp‘(i · (π / 2))) · (exp‘((1 / 2) · (log‘-𝐴)))))
71		efhalfpi 25981	. . . . . . . . 9 ⊢ (exp‘(i · (π / 2))) = i
72	71	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘(i · (π / 2))) = i)
73		negcl 11460	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
74	73	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℂ)
75	1	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (1 / 2) ∈ ℂ)
76		cxpef 26173	. . . . . . . . . 10 ⊢ ((-𝐴 ∈ ℂ ∧ -𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (-𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘-𝐴))))
77	74, 32, 75, 76	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘-𝐴))))
78		ax-1cn 11168	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
79		2halves 12440	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1 / 2) + (1 / 2)) = 1
81	80	oveq2i 7420	. . . . . . . . . . . 12 ⊢ (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = (-𝐴↑_𝑐1)
82		cxp1 26179	. . . . . . . . . . . . 13 ⊢ (-𝐴 ∈ ℂ → (-𝐴↑_𝑐1) = -𝐴)
83	74, 82	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐1) = -𝐴)
84	81, 83	eqtrid 2785	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = -𝐴)
85		rpcxpcl 26184	. . . . . . . . . . . . . . 15 ⊢ ((-𝐴 ∈ ℝ⁺ ∧ (1 / 2) ∈ ℝ) → (-𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
86	39, 2, 85	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
87	86	rpcnd 13018	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) ∈ ℂ)
88	87	sqvald 14108	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = ((-𝐴↑_𝑐(1 / 2)) · (-𝐴↑_𝑐(1 / 2))))
89		cxpadd 26187	. . . . . . . . . . . . 13 ⊢ (((-𝐴 ∈ ℂ ∧ -𝐴 ≠ 0) ∧ (1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((-𝐴↑_𝑐(1 / 2)) · (-𝐴↑_𝑐(1 / 2))))
90	74, 32, 75, 75, 89	syl211anc 1377	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((-𝐴↑_𝑐(1 / 2)) · (-𝐴↑_𝑐(1 / 2))))
91	88, 90	eqtr4d 2776	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = (-𝐴↑_𝑐((1 / 2) + (1 / 2))))
92	74	sqsqrtd 15386	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘-𝐴)↑2) = -𝐴)
93	84, 91, 92	3eqtr4d 2783	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = ((√‘-𝐴)↑2))
94	86	rprege0d 13023	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2)) ∈ ℝ ∧ 0 ≤ (-𝐴↑_𝑐(1 / 2))))
95	39	rpsqrtcld 15358	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘-𝐴) ∈ ℝ⁺)
96	95	rprege0d 13023	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘-𝐴) ∈ ℝ ∧ 0 ≤ (√‘-𝐴)))
97		sq11 14096	. . . . . . . . . . 11 ⊢ ((((-𝐴↑_𝑐(1 / 2)) ∈ ℝ ∧ 0 ≤ (-𝐴↑_𝑐(1 / 2))) ∧ ((√‘-𝐴) ∈ ℝ ∧ 0 ≤ (√‘-𝐴))) → (((-𝐴↑_𝑐(1 / 2))↑2) = ((√‘-𝐴)↑2) ↔ (-𝐴↑_𝑐(1 / 2)) = (√‘-𝐴)))
98	94, 96, 97	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (((-𝐴↑_𝑐(1 / 2))↑2) = ((√‘-𝐴)↑2) ↔ (-𝐴↑_𝑐(1 / 2)) = (√‘-𝐴)))
99	93, 98	mpbid 231	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) = (√‘-𝐴))
100	77, 99	eqtr3d 2775	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘-𝐴))) = (√‘-𝐴))
101	72, 100	oveq12d 7427	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((exp‘(i · (π / 2))) · (exp‘((1 / 2) · (log‘-𝐴)))) = (i · (√‘-𝐴)))
102	64, 70, 101	3eqtrd 2777	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (i · (√‘-𝐴)))
103		cxpef 26173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
104	1, 103	mp3an3 1451	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
105	104	adantr 482	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
106	43	fveq2d 6896	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘--𝐴) = (√‘𝐴))
107	39	rpge0d 13020	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -𝐴)
108	28, 107	sqrtnegd 15368	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘--𝐴) = (i · (√‘-𝐴)))
109	106, 108	eqtr3d 2775	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) = (i · (√‘-𝐴)))
110	102, 105, 109	3eqtr4d 2783	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
111	110	ex 414	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
112	80	oveq2i 7420	. . . . . . . . 9 ⊢ (𝐴↑_𝑐((1 / 2) + (1 / 2))) = (𝐴↑_𝑐1)
113		cxpadd 26187	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
114	1, 1, 113	mp3an23 1454	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
115		cxp1 26179	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐1) = 𝐴)
116	115	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐1) = 𝐴)
117	112, 114, 116	3eqtr3a 2797	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))) = 𝐴)
118		cxpcl 26182	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
119	1, 118	mpan2 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
120	119	sqvald 14108	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
121	120	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
122	20	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
123	117, 121, 122	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2))
124		sqeqor 14180	. . . . . . . . 9 ⊢ (((𝐴↑_𝑐(1 / 2)) ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → (((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2) ↔ ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))))
125	119, 14, 124	syl2anc 585	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2) ↔ ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))))
126	125	biimpa 478	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2)) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
127	123, 126	syldan 592	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
128	127	ord 863	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = (√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
129	128	con1d 145	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
130	111, 129	pm2.61d 179	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
131	130	ex 414	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
132	12, 131	pm2.61dne 3029	1 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 ici 11112 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 -cneg 11445 / cdiv 11871 2c2 12267 ℝ⁺crp 12974 ↑cexp 14027 √csqrt 15180 expce 16005 πcpi 16010 logclog 26063 ↑_𝑐ccxp 26064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066

This theorem is referenced by: logsqrt 26212 dvsqrt 26250 dvcnsqrt 26252 resqrtcn 26257 sqrtcn 26258 sqrt2cxp2logb9e3 26304 efiatan 26417 efiatan2 26422 sqrtlim 26477 chpchtlim 26982 logdivsqrle 33693

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