cxpsqrt - Metamath Proof Explorer

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Theorem cxpsqrt 27064

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 12590	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
2		halfre 12589	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
3		halfgt0 12591	. . . . . . 7 ⊢ 0 < (1 / 2)
4	2, 3	gt0ne0ii 11882	. . . . . 6 ⊢ (1 / 2) ≠ 0
5		0cxp 27027	. . . . . 6 ⊢ (((1 / 2) ∈ ℂ ∧ (1 / 2) ≠ 0) → (0↑_𝑐(1 / 2)) = 0)
6	1, 4, 5	mp2an 705	. . . . 5 ⊢ (0↑_𝑐(1 / 2)) = 0
7		sqrt0 15440	. . . . 5 ⊢ (√‘0) = 0
8	6, 7	eqtr4i 2789	. . . 4 ⊢ (0↑_𝑐(1 / 2)) = (√‘0)
9		oveq1 7453	. . . 4 ⊢ (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (0↑_𝑐(1 / 2)))
10		fveq2 6911	. . . 4 ⊢ (𝐴 = 0 → (√‘𝐴) = (√‘0))
11	8, 9, 10	3eqtr4a 2824	. . 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 11289	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
14		sqrtcl 15561	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
15	14	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
16		sqmul 14292	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
17	13, 15, 16	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = ((i↑2) · ((√‘𝐴)↑2)))
18		i2 14376	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
19	18	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i↑2) = -1)
20		sqrtth 15564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
21	20	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘𝐴)↑2) = 𝐴)
22	19, 21	oveq12d 7464	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i↑2) · ((√‘𝐴)↑2)) = ( -1 · 𝐴))
23		mulm1 11787	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ( -1 · 𝐴) = -𝐴)
24	23	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -1 · 𝐴) = -𝐴)
25	17, 22, 24	3eqtrd 2802	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = -𝐴)
26		cxpsqrtlem 27063	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)
27	26	resqcld 14298	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ∈ ℝ)
28	25, 27	eqeltrrd 2865	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ)
29		negeq0 11642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))
30	29	necon3bid 3003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))
31	30	biimpa 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -𝐴 ≠ 0)
32	31	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ≠ 0)
33	25, 32	eqnetrd 3026	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ≠ 0)
34		sq0i 14366	. . . . . . . . . . . . . . . . . 18 ⊢ ((i · (√‘𝐴)) = 0 → ((i · (√‘𝐴))↑2) = 0)
35	34	necon3i 2991	. . . . . . . . . . . . . . . . 17 ⊢ (((i · (√‘𝐴))↑2) ≠ 0 → (i · (√‘𝐴)) ≠ 0)
36	33, 35	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ≠ 0)
37	26, 36	sqgt0d 14424	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < ((i · (√‘𝐴))↑2))
38	37, 25	breqtrd 5135	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < -𝐴)
39	28, 38	elrpd 13191	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ⁺)
40		logneg 26949	. . . . . . . . . . . . 13 ⊢ ( -𝐴 ∈ ℝ⁺ → (log‘ - -𝐴) = ((log‘ -𝐴) + (i · π)))
41	39, 40	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ - -𝐴) = ((log‘ -𝐴) + (i · π)))
42		negneg 11638	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → - -𝐴 = 𝐴)
43	42	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → - -𝐴 = 𝐴)
44	43	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ - -𝐴) = (log‘𝐴))
45	39	relogcld 26984	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ -𝐴) ∈ ℝ)
46	45	recnd 11367	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ -𝐴) ∈ ℂ)
47		picn 26818	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
48	13, 47	mulcli 11346	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
49		addcom 11526	. . . . . . . . . . . . 13 ⊢ (((log‘ -𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘ -𝐴) + (i · π)) = ((i · π) + (log‘ -𝐴)))
50	46, 48, 49	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((log‘ -𝐴) + (i · π)) = ((i · π) + (log‘ -𝐴)))
51	41, 44, 50	3eqtr3d 2806	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘𝐴) = ((i · π) + (log‘ -𝐴)))
52	51	oveq2d 7462	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((1 / 2) · ((i · π) + (log‘ -𝐴))))
53		adddi 11319	. . . . . . . . . . 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 1495	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘ -𝐴))))
56		2cn 12448	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
57		2ne0 12479	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
58		divrec2 12021	. . . . . . . . . . . 12 ⊢ (((i · π) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((i · π) / 2) = ((1 / 2) · (i · π)))
59	48, 56, 57, 58	mp3an 1491	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = ((1 / 2) · (i · π))
60	13, 47, 56, 57	divassi 12103	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = (i · (π / 2))
61	59, 60	eqtr3i 2788	. . . . . . . . . 10 ⊢ ((1 / 2) · (i · π)) = (i · (π / 2))
62	61	oveq1i 7456	. . . . . . . . 9 ⊢ (((1 / 2) · (i · π)) + ((1 / 2) · (log‘ -𝐴))) = ((i · (π / 2)) + ((1 / 2) · (log‘ -𝐴)))
63	55, 62	eqtrdi 2814	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((i · (π / 2)) + ((1 / 2) · (log‘ -𝐴))))
64	63	fveq2d 6915	. . . . . . 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 12089	. . . . . . . . 9 ⊢ (π / 2) ∈ ℂ
66	13, 65	mulcli 11346	. . . . . . . 8 ⊢ (i · (π / 2)) ∈ ℂ
67		mulcl 11314	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘ -𝐴) ∈ ℂ) → ((1 / 2) · (log‘ -𝐴)) ∈ ℂ)
68	1, 46, 67	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘ -𝐴)) ∈ ℂ)
69		efadd 16292	. . . . . . . 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 599	. . . . . . 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 26833	. . . . . . . . 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 11587	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
74	73	ad2antrr 739	. . . . . . . . . 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 27026	. . . . . . . . . 10 ⊢ (( -𝐴 ∈ ℂ ∧ -𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → ( -𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘ -𝐴))))
77	74, 32, 75, 76	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘ -𝐴))))
78		ax-1cn 11288	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
79		2halves 12594	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1 / 2) + (1 / 2)) = 1
81	80	oveq2i 7457	. . . . . . . . . . . 12 ⊢ ( -𝐴↑_𝑐((1 / 2) + (1 / 2))) = ( -𝐴↑_𝑐1)
82		cxp1 27032	. . . . . . . . . . . . 13 ⊢ ( -𝐴 ∈ ℂ → ( -𝐴↑_𝑐1) = -𝐴)
83	74, 82	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐1) = -𝐴)
84	81, 83	eqtrid 2810	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐((1 / 2) + (1 / 2))) = -𝐴)
85		rpcxpcl 27037	. . . . . . . . . . . . . . 15 ⊢ (( -𝐴 ∈ ℝ⁺ ∧ (1 / 2) ∈ ℝ) → ( -𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
86	39, 2, 85	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) ∈ ℝ⁺)
87	86	rpcnd 13196	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) ∈ ℂ)
88	87	sqvald 14316	. . . . . . . . . . . 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 27040	. . . . . . . . . . . . 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 1404	. . . . . . . . . . . 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 2801	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2))↑2) = ( -𝐴↑_𝑐((1 / 2) + (1 / 2))))
92	74	sqsqrtd 15641	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘ -𝐴)↑2) = -𝐴)
93	84, 91, 92	3eqtr4d 2808	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2))↑2) = ((√‘ -𝐴)↑2))
94	86	rprege0d 13201	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2)) ∈ ℝ ∧ 0 ≤ ( -𝐴↑_𝑐(1 / 2))))
95	39	rpsqrtcld 15611	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ -𝐴) ∈ ℝ⁺)
96	95	rprege0d 13201	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘ -𝐴) ∈ ℝ ∧ 0 ≤ (√‘ -𝐴)))
97		sq11 14304	. . . . . . . . . . 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 596	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) = (√‘ -𝐴))
100	77, 99	eqtr3d 2800	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘ -𝐴))) = (√‘ -𝐴))
101	72, 100	oveq12d 7464	. . . . . . 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 2802	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (i · (√‘ -𝐴)))
103		cxpef 27026	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
104	1, 103	mp3an3 1480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
105	104	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
106	43	fveq2d 6915	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ - -𝐴) = (√‘𝐴))
107	39	rpge0d 13198	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -𝐴)
108	28, 107	sqrtnegd 15621	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ - -𝐴) = (i · (√‘ -𝐴)))
109	106, 108	eqtr3d 2800	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) = (i · (√‘ -𝐴)))
110	102, 105, 109	3eqtr4d 2808	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
111	110	ex 418	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
112	80	oveq2i 7457	. . . . . . . . 9 ⊢ (𝐴↑_𝑐((1 / 2) + (1 / 2))) = (𝐴↑_𝑐1)
113		cxpadd 27040	. . . . . . . . . 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 1483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
115		cxp1 27032	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐1) = 𝐴)
116	115	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐1) = 𝐴)
117	112, 114, 116	3eqtr3a 2822	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))) = 𝐴)
118		cxpcl 27035	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
119	1, 118	mpan2 704	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) ∈ ℂ)
120	119	sqvald 14316	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
121	120	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
122	20	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
123	117, 121, 122	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2))
124		sqeqor 14390	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2) ↔ ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))))
126	125	biimpa 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2)) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
127	123, 126	syldan 603	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) = (√‘𝐴) ∨ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
128	127	ord 878	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = (√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)))
129	128	con1d 146	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
130	111, 129	pm2.61d 181	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))
131	130	ex 418	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴)))
132	12, 131	pm2.61dne 3045	1 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐(1 / 2)) = (√‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 ‘cfv 6561 (class class class)co 7446 ℂcc 11228 ℝcr 11229 0cc0 11230 1c1 11231 ici 11232 + caddc 11233 · cmul 11235 < clt 11373 ≤ cle 11374 -cneg 11572 / cdiv 12003 2c2 12427 ℝ⁺crp 13150 ↑cexp 14234 √csqrt 15432 expce 16259 πcpi 16264 logclog 26915 ↑_𝑐ccxp 26916

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-pm 8881 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-ioc 13511 df-ico 13512 df-icc 13513 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-fac 14448 df-bc 14477 df-hash 14505 df-shft 15252 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-limsup 15670 df-clim 15687 df-rlim 15688 df-sum 15886 df-ef 16265 df-sin 16267 df-cos 16268 df-pi 16270 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-fbas 21708 df-fg 21709 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-ntr 23371 df-cls 23372 df-nei 23449 df-lp 23487 df-perf 23488 df-cn 23578 df-cnp 23579 df-haus 23666 df-tx 23914 df-hmeo 24107 df-fil 24198 df-fm 24290 df-flim 24291 df-flf 24292 df-xms 24672 df-ms 24673 df-tms 24674 df-cncf 25232 df-limc 26219 df-dv 26220 df-log 26917 df-cxp 26918

This theorem is used by: logsqrt 27065 dvsqrt 27103 dvcnsqrt 27105 resqrtcn 27110 sqrtcn 27111 sqrt2cxp2logb9e3 27160 efiatan 27273 efiatan2 27278 sqrtlim 27333 chpchtlim 27839 logdivsqrle 35362

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