cxpsqrt - Metamath Proof Explorer

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Theorem cxpsqrt 24249

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 11124	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
2		halfre 11123	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
3		halfgt0 11125	. . . . . . 7 ⊢ 0 < (1 / 2)
4	2, 3	gt0ne0ii 10443	. . . . . 6 ⊢ (1 / 2) ≠ 0
5		0cxp 24212	. . . . . 6 ⊢ (((1 / 2) ∈ ℂ ∧ (1 / 2) ≠ 0) → (0↑_𝑐(1 / 2)) = 0)
6	1, 4, 5	mp2an 704	. . . . 5 ⊢ (0↑_𝑐(1 / 2)) = 0
7		sqrt0 13830	. . . . 5 ⊢ (√‘0) = 0
8	6, 7	eqtr4i 2635	. . . 4 ⊢ (0↑_𝑐(1 / 2)) = (√‘0)
9		oveq1 6556	. . . 4 ⊢ (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (0↑_𝑐(1 / 2)))
10		fveq2 6103	. . . 4 ⊢ (𝐴 = 0 → (√‘𝐴) = (√‘0))
11	8, 9, 10	3eqtr4a 2670	. . 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 9874	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
14		sqrtcl 13949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
15	14	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
16		sqmul 12788	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
17	13, 15, 16	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
18		i2 12827	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
19	18	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i↑2) = -1)
20		sqrtth 13952	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
21	20	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘𝐴)↑2) = 𝐴)
22	19, 21	oveq12d 6567	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i↑2) · ((√‘𝐴)↑2)) = (-1 · 𝐴))
23		mulm1 10350	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴)
24	23	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-1 · 𝐴) = -𝐴)
25	17, 22, 24	3eqtrd 2648	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = -𝐴)
26		cxpsqrtlem 24248	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)
27	26	resqcld 12897	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ∈ ℝ)
28	25, 27	eqeltrrd 2689	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ)
29		negeq0 10214	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))
30	29	necon3bid 2826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))
31	30	biimpa 500	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -𝐴 ≠ 0)
32	31	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ≠ 0)
33	25, 32	eqnetrd 2849	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ≠ 0)
34		sq0i 12818	. . . . . . . . . . . . . . . . . 18 ⊢ ((i · (√‘𝐴)) = 0 → ((i · (√‘𝐴))↑2) = 0)
35	34	necon3i 2814	. . . . . . . . . . . . . . . . 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 12899	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < ((i · (√‘𝐴))↑2))
38	37, 25	breqtrd 4609	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < -𝐴)
39	28, 38	elrpd 11745	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ⁺)
40		logneg 24138	. . . . . . . . . . . . 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 10210	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
43	42	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → --𝐴 = 𝐴)
44	43	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘--𝐴) = (log‘𝐴))
45	39	relogcld 24173	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘-𝐴) ∈ ℝ)
46	45	recnd 9947	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘-𝐴) ∈ ℂ)
47		picn 24015	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
48	13, 47	mulcli 9924	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
49		addcom 10101	. . . . . . . . . . . . 13 ⊢ (((log‘-𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘-𝐴) + (i · π)) = ((i · π) + (log‘-𝐴)))
50	46, 48, 49	sylancl 693	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((log‘-𝐴) + (i · π)) = ((i · π) + (log‘-𝐴)))
51	41, 44, 50	3eqtr3d 2652	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘𝐴) = ((i · π) + (log‘-𝐴)))
52	51	oveq2d 6565	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((1 / 2) · ((i · π) + (log‘-𝐴))))
53		adddi 9904	. . . . . . . . . . . 12 ⊢ (((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, 53	mp3an12 1406	. . . . . . . . . . 11 ⊢ ((log‘-𝐴) ∈ ℂ → ((1 / 2) · ((i · π) + (log‘-𝐴))) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
55	46, 54	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · ((i · π) + (log‘-𝐴))) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
56	52, 55	eqtrd 2644	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))))
57		2cn 10968	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
58		2ne0 10990	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
59		divrec2 10581	. . . . . . . . . . . 12 ⊢ (((i · π) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((i · π) / 2) = ((1 / 2) · (i · π)))
60	48, 57, 58, 59	mp3an 1416	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = ((1 / 2) · (i · π))
61	13, 47, 57, 58	divassi 10660	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = (i · (π / 2))
62	60, 61	eqtr3i 2634	. . . . . . . . . 10 ⊢ ((1 / 2) · (i · π)) = (i · (π / 2))
63	62	oveq1i 6559	. . . . . . . . 9 ⊢ (((1 / 2) · (i · π)) + ((1 / 2) · (log‘-𝐴))) = ((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))
64	56, 63	syl6eq 2660	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((i · (π / 2)) + ((1 / 2) · (log‘-𝐴))))
65	64	fveq2d 6107	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (exp‘((i · (π / 2)) + ((1 / 2) · (log‘-𝐴)))))
66	47, 57, 58	divcli 10646	. . . . . . . . 9 ⊢ (π / 2) ∈ ℂ
67	13, 66	mulcli 9924	. . . . . . . 8 ⊢ (i · (π / 2)) ∈ ℂ
68		mulcl 9899	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘-𝐴) ∈ ℂ) → ((1 / 2) · (log‘-𝐴)) ∈ ℂ)
69	1, 46, 68	sylancr 694	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘-𝐴)) ∈ ℂ)
70		efadd 14663	. . . . . . . 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)))))
71	67, 69, 70	sylancr 694	. . . . . . 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)))))
72		efhalfpi 24027	. . . . . . . . 9 ⊢ (exp‘(i · (π / 2))) = i
73	72	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘(i · (π / 2))) = i)
74		negcl 10160	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
75	74	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℂ)
76	1	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (1 / 2) ∈ ℂ)
77		cxpef 24211	. . . . . . . . . 10 ⊢ ((-𝐴 ∈ ℂ ∧ -𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (-𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘-𝐴))))
78	75, 32, 76, 77	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘-𝐴))))
79		ax-1cn 9873	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
80		2halves 11137	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1)
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1 / 2) + (1 / 2)) = 1
82	81	oveq2i 6560	. . . . . . . . . . . 12 ⊢ (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = (-𝐴↑_𝑐1)
83		cxp1 24217	. . . . . . . . . . . . 13 ⊢ (-𝐴 ∈ ℂ → (-𝐴↑_𝑐1) = -𝐴)
84	75, 83	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐1) = -𝐴)
85	82, 84	syl5eq 2656	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = -𝐴)
86		rpcxpcl 24222	. . . . . . . . . . . . . . 15 ⊢ ((-𝐴 ∈ ℝ⁺ ∧ (1 / 2) ∈ ℝ) → (-𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
87	39, 2, 86	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
88	87	rpcnd 11750	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) ∈ ℂ)
89	88	sqvald 12867	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = ((-𝐴↑_𝑐(1 / 2)) · (-𝐴↑_𝑐(1 / 2))))
90		cxpadd 24225	. . . . . . . . . . . . 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))))
91	75, 32, 76, 76, 90	syl211anc 1324	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((-𝐴↑_𝑐(1 / 2)) · (-𝐴↑_𝑐(1 / 2))))
92	89, 91	eqtr4d 2647	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = (-𝐴↑_𝑐((1 / 2) + (1 / 2))))
93	75	sqsqrtd 14026	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘-𝐴)↑2) = -𝐴)
94	85, 92, 93	3eqtr4d 2654	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2))↑2) = ((√‘-𝐴)↑2))
95	87	rprege0d 11755	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((-𝐴↑_𝑐(1 / 2)) ∈ ℝ ∧ 0 ≤ (-𝐴↑_𝑐(1 / 2))))
96	39	rpsqrtcld 13998	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘-𝐴) ∈ ℝ⁺)
97	96	rprege0d 11755	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘-𝐴) ∈ ℝ ∧ 0 ≤ (√‘-𝐴)))
98		sq11 12798	. . . . . . . . . . 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)))
99	95, 97, 98	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (((-𝐴↑_𝑐(1 / 2))↑2) = ((√‘-𝐴)↑2) ↔ (-𝐴↑_𝑐(1 / 2)) = (√‘-𝐴)))
100	94, 99	mpbid 221	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (-𝐴↑_𝑐(1 / 2)) = (√‘-𝐴))
101	78, 100	eqtr3d 2646	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘-𝐴))) = (√‘-𝐴))
102	73, 101	oveq12d 6567	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((exp‘(i · (π / 2))) · (exp‘((1 / 2) · (log‘-𝐴)))) = (i · (√‘-𝐴)))
103	65, 71, 102	3eqtrd 2648	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (i · (√‘-𝐴)))
104		cxpef 24211	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
105	1, 104	mp3an3 1405	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
106	105	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
107	43	fveq2d 6107	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘--𝐴) = (√‘𝐴))
108	39	rpge0d 11752	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -𝐴)
109	28, 108	sqrtnegd 14008	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘--𝐴) = (i · (√‘-𝐴)))
110	107, 109	eqtr3d 2646	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) = (i · (√‘-𝐴)))
111	103, 106, 110	3eqtr4d 2654	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
112	111	ex 449	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
113	81	oveq2i 6560	. . . . . . . . 9 ⊢ (𝐴↑_𝑐((1 / 2) + (1 / 2))) = (𝐴↑_𝑐1)
114		cxpadd 24225	. . . . . . . . . 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))))
115	1, 1, 114	mp3an23 1408	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
116		cxp1 24217	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐1) = 𝐴)
117	116	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐1) = 𝐴)
118	113, 115, 117	3eqtr3a 2668	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))) = 𝐴)
119		cxpcl 24220	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
120	1, 119	mpan2 703	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
121	120	sqvald 12867	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
122	121	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
123	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
124	118, 122, 123	3eqtr4d 2654	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2))
125		sqeqor 12840	. . . . . . . . 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))))
126	120, 14, 125	syl2anc 691	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2) ↔ ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))))
127	126	biimpa 500	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2)) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
128	124, 127	syldan 486	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
129	128	ord 391	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = (√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
130	129	con1d 138	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
131	112, 130	pm2.61d 169	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
132	131	ex 449	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
133	12, 132	pm2.61dne 2868	1 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 -cneg 10146 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 ↑cexp 12722 √csqrt 13821 expce 14631 πcpi 14636 logclog 24105 ↑_𝑐ccxp 24106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108

This theorem is referenced by: logsqrt 24250 dvsqrt 24283 dvcnsqrt 24285 resqrtcn 24290 sqrtcn 24291 efiatan 24439 efiatan2 24444 sqrtlim 24499 chpchtlim 24968

W3C validator