cxpsqrt - Metamath Proof Explorer

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Theorem cxpsqrt 27031

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 12560	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
2		halfre 12559	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
3		halfgt0 12561	. . . . . . 7 ⊢ 0 < (1 / 2)
4	2, 3	gt0ne0ii 11852	. . . . . 6 ⊢ (1 / 2) ≠ 0
5		0cxp 26994	. . . . . 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 15408	. . . . 5 ⊢ (√‘0) = 0
8	6, 7	eqtr4i 2787	. . . 4 ⊢ (0↑_𝑐(1 / 2)) = (√‘0)
9		oveq1 7427	. . . 4 ⊢ (𝐴 = 0 → (𝐴↑_𝑐(1 / 2)) = (0↑_𝑐(1 / 2)))
10		fveq2 6885	. . . 4 ⊢ (𝐴 = 0 → (√‘𝐴) = (√‘0))
11	8, 9, 10	3eqtr4a 2822	. . 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 11259	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
14		sqrtcl 15529	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
15	14	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
16		sqmul 14262	. . . . . . . . . . . . . . . . 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 14346	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
19	18	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i↑2) = -1)
20		sqrtth 15532	. . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i↑2) · ((√‘𝐴)↑2)) = ( -1 · 𝐴))
23		mulm1 11757	. . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) = -𝐴)
26		cxpsqrtlem 27030	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)
27	26	resqcld 14268	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ∈ ℝ)
28	25, 27	eqeltrrd 2862	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ)
29		negeq0 11612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))
30	29	necon3bid 3000	. . . . . . . . . . . . . . . . . . . 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 3023	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴))↑2) ≠ 0)
34		sq0i 14336	. . . . . . . . . . . . . . . . . 18 ⊢ ((i · (√‘𝐴)) = 0 → ((i · (√‘𝐴))↑2) = 0)
35	34	necon3i 2988	. . . . . . . . . . . . . . . . 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 14394	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < ((i · (√‘𝐴))↑2))
38	37, 25	breqtrd 5131	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 < -𝐴)
39	28, 38	elrpd 13161	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → -𝐴 ∈ ℝ⁺)
40		logneg 26916	. . . . . . . . . . . . 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 11608	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → - -𝐴 = 𝐴)
43	42	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → - -𝐴 = 𝐴)
44	43	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ - -𝐴) = (log‘𝐴))
45	39	relogcld 26951	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ -𝐴) ∈ ℝ)
46	45	recnd 11337	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘ -𝐴) ∈ ℂ)
47		picn 26785	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
48	13, 47	mulcli 11316	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
49		addcom 11496	. . . . . . . . . . . . 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 2804	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (log‘𝐴) = ((i · π) + (log‘ -𝐴)))
52	51	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((1 / 2) · ((i · π) + (log‘ -𝐴))))
53		adddi 11289	. . . . . . . . . . 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 1494	. . . . . . . . . 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 2796	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = (((1 / 2) · (i · π)) + ((1 / 2) · (log‘ -𝐴))))
56		2cn 12418	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
57		2ne0 12449	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
58		divrec2 11991	. . . . . . . . . . . 12 ⊢ (((i · π) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((i · π) / 2) = ((1 / 2) · (i · π)))
59	48, 56, 57, 58	mp3an 1490	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = ((1 / 2) · (i · π))
60	13, 47, 56, 57	divassi 12073	. . . . . . . . . . 11 ⊢ ((i · π) / 2) = (i · (π / 2))
61	59, 60	eqtr3i 2786	. . . . . . . . . 10 ⊢ ((1 / 2) · (i · π)) = (i · (π / 2))
62	61	oveq1i 7430	. . . . . . . . 9 ⊢ (((1 / 2) · (i · π)) + ((1 / 2) · (log‘ -𝐴))) = ((i · (π / 2)) + ((1 / 2) · (log‘ -𝐴)))
63	55, 62	eqtrdi 2812	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((1 / 2) · (log‘𝐴)) = ((i · (π / 2)) + ((1 / 2) · (log‘ -𝐴))))
64	63	fveq2d 6889	. . . . . . 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 12059	. . . . . . . . 9 ⊢ (π / 2) ∈ ℂ
66	13, 65	mulcli 11316	. . . . . . . 8 ⊢ (i · (π / 2)) ∈ ℂ
67		mulcl 11284	. . . . . . . . 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 16260	. . . . . . . 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 26800	. . . . . . . . 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 11557	. . . . . . . . . . 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 26993	. . . . . . . . . 10 ⊢ (( -𝐴 ∈ ℂ ∧ -𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → ( -𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘ -𝐴))))
77	74, 32, 75, 76	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘ -𝐴))))
78		ax-1cn 11258	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
79		2halves 12564	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1 / 2) + (1 / 2)) = 1
81	80	oveq2i 7431	. . . . . . . . . . . 12 ⊢ ( -𝐴↑_𝑐((1 / 2) + (1 / 2))) = ( -𝐴↑_𝑐1)
82		cxp1 26999	. . . . . . . . . . . . 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 2808	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐((1 / 2) + (1 / 2))) = -𝐴)
85		rpcxpcl 27004	. . . . . . . . . . . . . . 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 13166	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ( -𝐴↑_𝑐(1 / 2)) ∈ ℂ)
88	87	sqvald 14286	. . . . . . . . . . . 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 27007	. . . . . . . . . . . . 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 1403	. . . . . . . . . . . 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 2799	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2))↑2) = ( -𝐴↑_𝑐((1 / 2) + (1 / 2))))
92	74	sqsqrtd 15609	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘ -𝐴)↑2) = -𝐴)
93	84, 91, 92	3eqtr4d 2806	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2))↑2) = ((√‘ -𝐴)↑2))
94	86	rprege0d 13171	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (( -𝐴↑_𝑐(1 / 2)) ∈ ℝ ∧ 0 ≤ ( -𝐴↑_𝑐(1 / 2))))
95	39	rpsqrtcld 15579	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ -𝐴) ∈ ℝ⁺)
96	95	rprege0d 13171	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((√‘ -𝐴) ∈ ℝ ∧ 0 ≤ (√‘ -𝐴)))
97		sq11 14274	. . . . . . . . . . 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 2798	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘ -𝐴))) = (√‘ -𝐴))
101	72, 100	oveq12d 7438	. . . . . . 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 2800	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (exp‘((1 / 2) · (log‘𝐴))) = (i · (√‘ -𝐴)))
103		cxpef 26993	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
104	1, 103	mp3an3 1479	. . . . . . 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 6889	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ - -𝐴) = (√‘𝐴))
107	39	rpge0d 13168	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -𝐴)
108	28, 107	sqrtnegd 15589	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘ - -𝐴) = (i · (√‘ -𝐴)))
109	106, 108	eqtr3d 2798	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) = (i · (√‘ -𝐴)))
110	102, 105, 109	3eqtr4d 2806	. . . . 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 7431	. . . . . . . . 9 ⊢ (𝐴↑_𝑐((1 / 2) + (1 / 2))) = (𝐴↑_𝑐1)
113		cxpadd 27007	. . . . . . . . . 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 1482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐((1 / 2) + (1 / 2))) = ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))))
115		cxp1 26999	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑_𝑐1) = 𝐴)
116	115	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐1) = 𝐴)
117	112, 114, 116	3eqtr3a 2820	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2)) · (𝐴↑_𝑐(1 / 2))) = 𝐴)
118		cxpcl 27002	. . . . . . . . . . 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 14286	. . . . . . . . 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 2806	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑_𝑐(1 / 2))↑2) = ((√‘𝐴)↑2))
124		sqeqor 14360	. . . . . . . . 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 3042	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 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 ici 11202 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 -cneg 11542 / cdiv 11973 2c2 12397 ℝ⁺crp 13120 ↑cexp 14204 √csqrt 15400 expce 16227 πcpi 16232 logclog 26882 ↑_𝑐ccxp 26883

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884 df-cxp 26885

This theorem is used by: logsqrt 27032 dvsqrt 27070 dvcnsqrt 27072 resqrtcn 27077 sqrtcn 27078 sqrt2cxp2logb9e3 27127 efiatan 27240 efiatan2 27245 sqrtlim 27300 chpchtlim 27806 logdivsqrle 35279

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