Theorem cxpsqrt 23648

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 10829	. . . . . 6 |- ( 1 / 2 ) e. CC
2		halfre 10828	. . . . . . 7 |- ( 1 / 2 ) e. RR
3		halfgt0 10830	. . . . . . 7 |- 0 < ( 1 / 2 )
4	2, 3	gt0ne0ii 10150	. . . . . 6 |- ( 1 / 2 ) =/= 0
5		0cxp 23611	. . . . . 6 |- ( ( ( 1 / 2 ) e. CC /\ ( 1 / 2 ) =/= 0 ) -> ( 0 ^c ( 1 / 2 ) ) = 0 )
6	1, 4, 5	mp2an 678	. . . . 5 |- ( 0 ^c ( 1 / 2 ) ) = 0
7		sqrt0 13305	. . . . 5 |- ( sqr ` 0 ) = 0
8	6, 7	eqtr4i 2476	. . . 4 |- ( 0 ^c ( 1 / 2 ) ) = ( sqr ` 0 )
9		oveq1 6297	. . . 4 |- ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( 0 ^c ( 1 / 2 ) ) )
10		fveq2 5865	. . . 4 |- ( A = 0 -> ( sqr ` A ) = ( sqr ` 0 ) )
11	8, 9, 10	3eqtr4a 2511	. . 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 9598	. . . . . . . . . . . . . . . . 17 |- _i e. CC
14		sqrtcl 13424	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( sqr ` A ) e. CC )
15	14	ad2antrr 732	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` A ) e. CC )
16		sqmul 12338	. . . . . . . . . . . . . . . . 17 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
17	13, 15, 16	sylancr 669	. . . . . . . . . . . . . . . 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 12375	. . . . . . . . . . . . . . . . . 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 13427	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( ( sqr ` A ) ^ 2 ) = A )
21	20	ad2antrr 732	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
22	19, 21	oveq12d 6308	. . . . . . . . . . . . . . . 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 10060	. . . . . . . . . . . . . . . . 17 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
24	23	ad2antrr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u 1 x. A ) = -u A )
25	17, 22, 24	3eqtrd 2489	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A )
26		cxpsqrtlem 23647	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) e. RR )
27	26	resqcld 12442	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) e. RR )
28	25, 27	eqeltrrd 2530	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR )
29		negeq0 9928	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
30	29	necon3bid 2668	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) )
31	30	biimpa 487	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 )
32	31	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A =/= 0 )
33	25, 32	eqnetrd 2691	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 )
34		sq0i 12367	. . . . . . . . . . . . . . . . . 18 |- ( ( _i x. ( sqr ` A ) ) = 0 -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = 0 )
35	34	necon3i 2656	. . . . . . . . . . . . . . . . 17 |- ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 -> ( _i x. ( sqr ` A ) ) =/= 0 )
36	33, 35	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) =/= 0 )
37	26, 36	sqgt0d 12444	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < ( ( _i x. ( sqr ` A ) ) ^ 2 ) )
38	37, 25	breqtrd 4427	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < -u A )
39	28, 38	elrpd 11338	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR+ )
40		logneg 23537	. . . . . . . . . . . . 13 |- ( -u A e. RR+ -> ( log ` -u -u A ) = ( ( log ` -u A ) + ( _i x. pi ) ) )
41	39, 40	syl 17	. . . . . . . . . . . 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 9924	. . . . . . . . . . . . . 14 |- ( A e. CC -> -u -u A = A )
43	42	ad2antrr 732	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u -u A = A )
44	43	fveq2d 5869	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u -u A ) = ( log ` A ) )
45	39	relogcld 23572	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. RR )
46	45	recnd 9669	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. CC )
47		picn 23414	. . . . . . . . . . . . . 14 |- pi e. CC
48	13, 47	mulcli 9648	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
49		addcom 9819	. . . . . . . . . . . . 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 668	. . . . . . . . . . . 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 2493	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` A ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
52	51	oveq2d 6306	. . . . . . . . . 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 9628	. . . . . . . . . . . 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 1354	. . . . . . . . . . 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 17	. . . . . . . . . 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 2485	. . . . . . . . 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 10680	. . . . . . . . . . . 12 |- 2 e. CC
58		2ne0 10702	. . . . . . . . . . . 12 |- 2 =/= 0
59		divrec2 10287	. . . . . . . . . . . 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 1364	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( ( 1 / 2 ) x. ( _i x. pi ) )
61	13, 47, 57, 58	divassi 10363	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( _i x. ( pi / 2 ) )
62	60, 61	eqtr3i 2475	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( _i x. pi ) ) = ( _i x. ( pi / 2 ) )
63	62	oveq1i 6300	. . . . . . . . 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 2501	. . . . . . . 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 5869	. . . . . . 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 10349	. . . . . . . . 9 |- ( pi / 2 ) e. CC
67	13, 66	mulcli 9648	. . . . . . . 8 |- ( _i x. ( pi / 2 ) ) e. CC
68		mulcl 9623	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` -u A ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
69	1, 46, 68	sylancr 669	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
70		efadd 14148	. . . . . . . 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 669	. . . . . . 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 23426	. . . . . . . . 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 9875	. . . . . . . . . . 11 |- ( A e. CC -> -u A e. CC )
75	74	ad2antrr 732	. . . . . . . . . 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 23610	. . . . . . . . . 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 1268	. . . . . . . . 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 9597	. . . . . . . . . . . . . 14 |- 1 e. CC
80		2halves 10841	. . . . . . . . . . . . . 14 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
82	81	oveq2i 6301	. . . . . . . . . . . 12 |- ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( -u A ^c 1 )
83		cxp1 23616	. . . . . . . . . . . . 13 |- ( -u A e. CC -> ( -u A ^c 1 ) = -u A )
84	75, 83	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c 1 ) = -u A )
85	82, 84	syl5eq 2497	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = -u A )
86		rpcxpcl 23621	. . . . . . . . . . . . . . 15 |- ( ( -u A e. RR+ /\ ( 1 / 2 ) e. RR ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
87	39, 2, 86	sylancl 668	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
88	87	rpcnd 11343	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. CC )
89	88	sqvald 12413	. . . . . . . . . . . 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 23624	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 2488	. . . . . . . . . . 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 13501	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) ^ 2 ) = -u A )
94	85, 92, 93	3eqtr4d 2495	. . . . . . . . . 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 11348	. . . . . . . . . . 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 13473	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u A ) e. RR+ )
97	96	rprege0d 11348	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) e. RR /\ 0 <_ ( sqr ` -u A ) ) )
98		sq11 12347	. . . . . . . . . . 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 667	. . . . . . . . . 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 214	. . . . . . . . 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 2487	. . . . . . . 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 6308	. . . . . . 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 2489	. . . . . 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 23610	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
105	1, 104	mp3an3 1353	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
106	105	adantr 467	. . . . . 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 5869	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u -u A ) = ( sqr ` A ) )
108	39	rpge0d 11345	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 <_ -u A )
109	28, 108	sqrtnegd 13483	. . . . . . 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 2487	. . . . . 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 2495	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
112	111	ex 436	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
113	81	oveq2i 6301	. . . . . . . . 9 |- ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( A ^c 1 )
114		cxpadd 23624	. . . . . . . . . 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 1356	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
116		cxp1 23616	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c 1 ) = A )
117	116	adantr 467	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c 1 ) = A )
118	113, 115, 117	3eqtr3a 2509	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) = A )
119		cxpcl 23619	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) e. CC )
120	1, 119	mpan2 677	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) e. CC )
121	120	sqvald 12413	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
122	121	adantr 467	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
123	20	adantr 467	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( sqr ` A ) ^ 2 ) = A )
124	118, 122, 123	3eqtr4d 2495	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) )
125		sqeqor 12388	. . . . . . . . 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 667	. . . . . . . 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 487	. . . . . . 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 473	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
129	128	ord 379	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
130	129	con1d 128	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
131	112, 130	pm2.61d 162	. . 3 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
132	131	ex 436	. 2 |- ( A e. CC -> ( A =/= 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
133	12, 132	pm2.61dne 2710	1 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   _ici 9541   + caddc 9542   x. cmul 9544   < clt 9675   <_ cle 9676   -ucneg 9861   / cdiv 10269   2c2 10659   RR+crp 11302   ^cexp 12272   sqrcsqrt 13296   expce 14114   picpi 14119   logclog 23504   ^c ccxp 23505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507

This theorem is referenced by:  logsqrt  23649  dvsqrt  23682  dvcnsqrt  23684  resqrtcn  23689  sqrtcn  23690  efiatan  23838  efiatan2  23843  sqrtlim  23898  chpchtlim  24317