Theorem cxpsqrt 22950

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 e. CC -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )

Proof of Theorem cxpsqrt

Step	Hyp	Ref	Expression
1		halfcn 10767	. . . . . 6 |- ( 1 / 2 ) e. CC
2		halfre 10766	. . . . . . 7 |- ( 1 / 2 ) e. RR
3		halfgt0 10768	. . . . . . 7 |- 0 < ( 1 / 2 )
4	2, 3	gt0ne0ii 10101	. . . . . 6 |- ( 1 / 2 ) =/= 0
5		0cxp 22913	. . . . . 6 |- ( ( ( 1 / 2 ) e. CC /\ ( 1 / 2 ) =/= 0 ) -> ( 0 ^c ( 1 / 2 ) ) = 0 )
6	1, 4, 5	mp2an 672	. . . . 5 |- ( 0 ^c ( 1 / 2 ) ) = 0
7		sqrt0 13055	. . . . 5 |- ( sqr ` 0 ) = 0
8	6, 7	eqtr4i 2499	. . . 4 |- ( 0 ^c ( 1 / 2 ) ) = ( sqr ` 0 )
9		oveq1 6302	. . . 4 |- ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( 0 ^c ( 1 / 2 ) ) )
10		fveq2 5872	. . . 4 |- ( A = 0 -> ( sqr ` A ) = ( sqr ` 0 ) )
11	8, 9, 10	3eqtr4a 2534	. . 3 |- ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
12	11	a1i 11	. 2 |- ( A e. CC -> ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
13		ax-icn 9563	. . . . . . . . . . . . . . . . 17 |- _i e. CC
14		sqrtcl 13174	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( sqr ` A ) e. CC )
15	14	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` A ) e. CC )
16		sqmul 12211	. . . . . . . . . . . . . . . . 17 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
17	13, 15, 16	sylancr 663	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
18		i2 12248	. . . . . . . . . . . . . . . . . 18 |- ( _i ^ 2 ) = -u 1
19	18	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i ^ 2 ) = -u 1 )
20		sqrtth 13177	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( ( sqr ` A ) ^ 2 ) = A )
21	20	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
22	19, 21	oveq12d 6313	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) = ( -u 1 x. A ) )
23		mulm1 10010	. . . . . . . . . . . . . . . . 17 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
24	23	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u 1 x. A ) = -u A )
25	17, 22, 24	3eqtrd 2512	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A )
26		cxpsqrtlem 22949	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) e. RR )
27	26	resqcld 12316	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) e. RR )
28	25, 27	eqeltrrd 2556	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR )
29		negeq0 9885	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
30	29	necon3bid 2725	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) )
31	30	biimpa 484	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 )
32	31	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A =/= 0 )
33	25, 32	eqnetrd 2760	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 )
34		sq0i 12240	. . . . . . . . . . . . . . . . . 18 |- ( ( _i x. ( sqr ` A ) ) = 0 -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = 0 )
35	34	necon3i 2707	. . . . . . . . . . . . . . . . 17 |- ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 -> ( _i x. ( sqr ` A ) ) =/= 0 )
36	33, 35	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) =/= 0 )
37	26, 36	sqgt0d 12318	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < ( ( _i x. ( sqr ` A ) ) ^ 2 ) )
38	37, 25	breqtrd 4477	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < -u A )
39	28, 38	elrpd 11266	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR+ )
40		logneg 22838	. . . . . . . . . . . . 13 |- ( -u A e. RR+ -> ( log ` -u -u A ) = ( ( log ` -u A ) + ( _i x. pi ) ) )
41	39, 40	syl 16	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u -u A ) = ( ( log ` -u A ) + ( _i x. pi ) ) )
42		negneg 9881	. . . . . . . . . . . . . 14 |- ( A e. CC -> -u -u A = A )
43	42	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u -u A = A )
44	43	fveq2d 5876	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u -u A ) = ( log ` A ) )
45	39	relogcld 22874	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. RR )
46	45	recnd 9634	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. CC )
47		picn 22719	. . . . . . . . . . . . . 14 |- pi e. CC
48	13, 47	mulcli 9613	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
49		addcom 9777	. . . . . . . . . . . . 13 |- ( ( ( log ` -u A ) e. CC /\ ( _i x. pi ) e. CC ) -> ( ( log ` -u A ) + ( _i x. pi ) ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
50	46, 48, 49	sylancl 662	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( log ` -u A ) + ( _i x. pi ) ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
51	41, 44, 50	3eqtr3d 2516	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` A ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
52	51	oveq2d 6311	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` A ) ) = ( ( 1 / 2 ) x. ( ( _i x. pi ) + ( log ` -u A ) ) ) )
53		adddi 9593	. . . . . . . . . . . 12 |- ( ( ( 1 / 2 ) e. CC /\ ( _i x. pi ) e. CC /\ ( log ` -u A ) e. CC ) -> ( ( 1 / 2 ) x. ( ( _i x. pi ) + ( log ` -u A ) ) ) = ( ( ( 1 / 2 ) x. ( _i x. pi ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
54	1, 48, 53	mp3an12 1314	. . . . . . . . . . 11 |- ( ( log ` -u A ) e. CC -> ( ( 1 / 2 ) x. ( ( _i x. pi ) + ( log ` -u A ) ) ) = ( ( ( 1 / 2 ) x. ( _i x. pi ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
55	46, 54	syl 16	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( ( _i x. pi ) + ( log ` -u A ) ) ) = ( ( ( 1 / 2 ) x. ( _i x. pi ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
56	52, 55	eqtrd 2508	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` A ) ) = ( ( ( 1 / 2 ) x. ( _i x. pi ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
57		2cn 10618	. . . . . . . . . . . 12 |- 2 e. CC
58		2ne0 10640	. . . . . . . . . . . 12 |- 2 =/= 0
59		divrec2 10236	. . . . . . . . . . . 12 |- ( ( ( _i x. pi ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( _i x. pi ) / 2 ) = ( ( 1 / 2 ) x. ( _i x. pi ) ) )
60	48, 57, 58, 59	mp3an 1324	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( ( 1 / 2 ) x. ( _i x. pi ) )
61	13, 47, 57, 58	divassi 10312	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( _i x. ( pi / 2 ) )
62	60, 61	eqtr3i 2498	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( _i x. pi ) ) = ( _i x. ( pi / 2 ) )
63	62	oveq1i 6305	. . . . . . . . 9 |- ( ( ( 1 / 2 ) x. ( _i x. pi ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) = ( ( _i x. ( pi / 2 ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) )
64	56, 63	syl6eq 2524	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` A ) ) = ( ( _i x. ( pi / 2 ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
65	64	fveq2d 5876	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) = ( exp ` ( ( _i x. ( pi / 2 ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) )
66	47, 57, 58	divcli 10298	. . . . . . . . 9 |- ( pi / 2 ) e. CC
67	13, 66	mulcli 9613	. . . . . . . 8 |- ( _i x. ( pi / 2 ) ) e. CC
68		mulcl 9588	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` -u A ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
69	1, 46, 68	sylancr 663	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
70		efadd 13708	. . . . . . . 8 |- ( ( ( _i x. ( pi / 2 ) ) e. CC /\ ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC ) -> ( exp ` ( ( _i x. ( pi / 2 ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) = ( ( exp ` ( _i x. ( pi / 2 ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) )
71	67, 69, 70	sylancr 663	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( exp ` ( ( _i x. ( pi / 2 ) ) + ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) = ( ( exp ` ( _i x. ( pi / 2 ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) )
72		efhalfpi 22730	. . . . . . . . 9 |- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
73	72	a1i 11	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( exp ` ( _i x. ( pi / 2 ) ) ) = _i )
74		negcl 9832	. . . . . . . . . . 11 |- ( A e. CC -> -u A e. CC )
75	74	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. CC )
76	1	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( 1 / 2 ) e. CC )
77		cxpef 22912	. . . . . . . . . 10 |- ( ( -u A e. CC /\ -u A =/= 0 /\ ( 1 / 2 ) e. CC ) -> ( -u A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
78	75, 32, 76, 77	syl3anc 1228	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) )
79		ax-1cn 9562	. . . . . . . . . . . . . 14 |- 1 e. CC
80		2halves 10779	. . . . . . . . . . . . . 14 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
82	81	oveq2i 6306	. . . . . . . . . . . 12 |- ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( -u A ^c 1 )
83		cxp1 22918	. . . . . . . . . . . . 13 |- ( -u A e. CC -> ( -u A ^c 1 ) = -u A )
84	75, 83	syl 16	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c 1 ) = -u A )
85	82, 84	syl5eq 2520	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = -u A )
86		rpcxpcl 22923	. . . . . . . . . . . . . . 15 |- ( ( -u A e. RR+ /\ ( 1 / 2 ) e. RR ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
87	39, 2, 86	sylancl 662	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
88	87	rpcnd 11270	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. CC )
89	88	sqvald 12287	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( -u A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( -u A ^c ( 1 / 2 ) ) x. ( -u A ^c ( 1 / 2 ) ) ) )
90		cxpadd 22926	. . . . . . . . . . . . 13 |- ( ( ( -u A e. CC /\ -u A =/= 0 ) /\ ( 1 / 2 ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( -u A ^c ( 1 / 2 ) ) x. ( -u A ^c ( 1 / 2 ) ) ) )
91	75, 32, 76, 76, 90	syl211anc 1234	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( -u A ^c ( 1 / 2 ) ) x. ( -u A ^c ( 1 / 2 ) ) ) )
92	89, 91	eqtr4d 2511	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( -u A ^c ( 1 / 2 ) ) ^ 2 ) = ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) )
93	75	sqsqrtd 13250	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) ^ 2 ) = -u A )
94	85, 92, 93	3eqtr4d 2518	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( -u A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` -u A ) ^ 2 ) )
95	87	rprege0d 11275	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( -u A ^c ( 1 / 2 ) ) e. RR /\ 0 <_ ( -u A ^c ( 1 / 2 ) ) ) )
96	39	rpsqrtcld 13223	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u A ) e. RR+ )
97	96	rprege0d 11275	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) e. RR /\ 0 <_ ( sqr ` -u A ) ) )
98		sq11 12220	. . . . . . . . . . 11 |- ( ( ( ( -u A ^c ( 1 / 2 ) ) e. RR /\ 0 <_ ( -u A ^c ( 1 / 2 ) ) ) /\ ( ( sqr ` -u A ) e. RR /\ 0 <_ ( sqr ` -u A ) ) ) -> ( ( ( -u A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` -u A ) ^ 2 ) <-> ( -u A ^c ( 1 / 2 ) ) = ( sqr ` -u A ) ) )
99	95, 97, 98	syl2anc 661	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( ( -u A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` -u A ) ^ 2 ) <-> ( -u A ^c ( 1 / 2 ) ) = ( sqr ` -u A ) ) )
100	94, 99	mpbid 210	. . . . . . . . 9 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) = ( sqr ` -u A ) )
101	78, 100	eqtr3d 2510	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) = ( sqr ` -u A ) )
102	73, 101	oveq12d 6313	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( exp ` ( _i x. ( pi / 2 ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` -u A ) ) ) ) = ( _i x. ( sqr ` -u A ) ) )
103	65, 71, 102	3eqtrd 2512	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) = ( _i x. ( sqr ` -u A ) ) )
104		cxpef 22912	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
105	1, 104	mp3an3 1313	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
106	105	adantr 465	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
107	43	fveq2d 5876	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u -u A ) = ( sqr ` A ) )
108	39	rpge0d 11272	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 <_ -u A )
109	28, 108	sqrtnegd 13233	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u -u A ) = ( _i x. ( sqr ` -u A ) ) )
110	107, 109	eqtr3d 2510	. . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` A ) = ( _i x. ( sqr ` -u A ) ) )
111	103, 106, 110	3eqtr4d 2518	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
112	111	ex 434	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
113	81	oveq2i 6306	. . . . . . . . 9 |- ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( A ^c 1 )
114		cxpadd 22926	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( 1 / 2 ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
115	1, 1, 114	mp3an23 1316	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
116		cxp1 22918	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c 1 ) = A )
117	116	adantr 465	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c 1 ) = A )
118	113, 115, 117	3eqtr3a 2532	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) = A )
119		cxpcl 22921	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) e. CC )
120	1, 119	mpan2 671	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) e. CC )
121	120	sqvald 12287	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
122	121	adantr 465	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
123	20	adantr 465	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( sqr ` A ) ^ 2 ) = A )
124	118, 122, 123	3eqtr4d 2518	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) )
125		sqeqor 12262	. . . . . . . . 9 |- ( ( ( A ^c ( 1 / 2 ) ) e. CC /\ ( sqr ` A ) e. CC ) -> ( ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) <-> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) ) )
126	120, 14, 125	syl2anc 661	. . . . . . . 8 |- ( A e. CC -> ( ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) <-> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) ) )
127	126	biimpa 484	. . . . . . 7 |- ( ( A e. CC /\ ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) ) -> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
128	124, 127	syldan 470	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
129	128	ord 377	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
130	129	con1d 124	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
131	112, 130	pm2.61d 158	. . 3 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
132	131	ex 434	. 2 |- ( A e. CC -> ( A =/= 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
133	12, 132	pm2.61dne 2784	1 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505   _ici 9506   + caddc 9507   x. cmul 9509   < clt 9640   <_ cle 9641   -ucneg 9818   / cdiv 10218   2c2 10597   RR+crp 11232   ^cexp 12146   sqrcsqrt 13046   expce 13676   picpi 13681   logclog 22808   ^c ccxp 22809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810  df-cxp 22811

This theorem is referenced by:  logsqrt  22951  dvsqrt  22984  resqrtcn  22989  sqrtcn  22990  efiatan  23109  efiatan2  23114  sqrtlim  23168  chpchtlim  23530  dvcnsqrt  30028