Theorem cxpsqrt 23252

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 10751	. . . . . 6 |- ( 1 / 2 ) e. CC
2		halfre 10750	. . . . . . 7 |- ( 1 / 2 ) e. RR
3		halfgt0 10752	. . . . . . 7 |- 0 < ( 1 / 2 )
4	2, 3	gt0ne0ii 10085	. . . . . 6 |- ( 1 / 2 ) =/= 0
5		0cxp 23215	. . . . . 6 |- ( ( ( 1 / 2 ) e. CC /\ ( 1 / 2 ) =/= 0 ) -> ( 0 ^c ( 1 / 2 ) ) = 0 )
6	1, 4, 5	mp2an 670	. . . . 5 |- ( 0 ^c ( 1 / 2 ) ) = 0
7		sqrt0 13157	. . . . 5 |- ( sqr ` 0 ) = 0
8	6, 7	eqtr4i 2486	. . . 4 |- ( 0 ^c ( 1 / 2 ) ) = ( sqr ` 0 )
9		oveq1 6277	. . . 4 |- ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( 0 ^c ( 1 / 2 ) ) )
10		fveq2 5848	. . . 4 |- ( A = 0 -> ( sqr ` A ) = ( sqr ` 0 ) )
11	8, 9, 10	3eqtr4a 2521	. . 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 9540	. . . . . . . . . . . . . . . . 17 |- _i e. CC
14		sqrtcl 13276	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( sqr ` A ) e. CC )
15	14	ad2antrr 723	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` A ) e. CC )
16		sqmul 12213	. . . . . . . . . . . . . . . . 17 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
17	13, 15, 16	sylancr 661	. . . . . . . . . . . . . . . 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 12250	. . . . . . . . . . . . . . . . . 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 13279	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( ( sqr ` A ) ^ 2 ) = A )
21	20	ad2antrr 723	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
22	19, 21	oveq12d 6288	. . . . . . . . . . . . . . . 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 9994	. . . . . . . . . . . . . . . . 17 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
24	23	ad2antrr 723	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u 1 x. A ) = -u A )
25	17, 22, 24	3eqtrd 2499	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A )
26		cxpsqrtlem 23251	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) e. RR )
27	26	resqcld 12318	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) e. RR )
28	25, 27	eqeltrrd 2543	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR )
29		negeq0 9864	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
30	29	necon3bid 2712	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) )
31	30	biimpa 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 )
32	31	adantr 463	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A =/= 0 )
33	25, 32	eqnetrd 2747	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 )
34		sq0i 12242	. . . . . . . . . . . . . . . . . 18 |- ( ( _i x. ( sqr ` A ) ) = 0 -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = 0 )
35	34	necon3i 2694	. . . . . . . . . . . . . . . . 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 12320	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < ( ( _i x. ( sqr ` A ) ) ^ 2 ) )
38	37, 25	breqtrd 4463	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < -u A )
39	28, 38	elrpd 11256	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR+ )
40		logneg 23141	. . . . . . . . . . . . 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 9860	. . . . . . . . . . . . . 14 |- ( A e. CC -> -u -u A = A )
43	42	ad2antrr 723	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u -u A = A )
44	43	fveq2d 5852	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u -u A ) = ( log ` A ) )
45	39	relogcld 23176	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. RR )
46	45	recnd 9611	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. CC )
47		picn 23018	. . . . . . . . . . . . . 14 |- pi e. CC
48	13, 47	mulcli 9590	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
49		addcom 9755	. . . . . . . . . . . . 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 660	. . . . . . . . . . . 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 2503	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` A ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
52	51	oveq2d 6286	. . . . . . . . . 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 9570	. . . . . . . . . . . 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 1312	. . . . . . . . . . 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 2495	. . . . . . . . 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 10602	. . . . . . . . . . . 12 |- 2 e. CC
58		2ne0 10624	. . . . . . . . . . . 12 |- 2 =/= 0
59		divrec2 10220	. . . . . . . . . . . 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 1322	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( ( 1 / 2 ) x. ( _i x. pi ) )
61	13, 47, 57, 58	divassi 10296	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( _i x. ( pi / 2 ) )
62	60, 61	eqtr3i 2485	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( _i x. pi ) ) = ( _i x. ( pi / 2 ) )
63	62	oveq1i 6280	. . . . . . . . 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 2511	. . . . . . . 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 5852	. . . . . . 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 10282	. . . . . . . . 9 |- ( pi / 2 ) e. CC
67	13, 66	mulcli 9590	. . . . . . . 8 |- ( _i x. ( pi / 2 ) ) e. CC
68		mulcl 9565	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` -u A ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
69	1, 46, 68	sylancr 661	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
70		efadd 13911	. . . . . . . 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 661	. . . . . . 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 23030	. . . . . . . . 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 9811	. . . . . . . . . . 11 |- ( A e. CC -> -u A e. CC )
75	74	ad2antrr 723	. . . . . . . . . 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 23214	. . . . . . . . . 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 1226	. . . . . . . . 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 9539	. . . . . . . . . . . . . 14 |- 1 e. CC
80		2halves 10763	. . . . . . . . . . . . . 14 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
82	81	oveq2i 6281	. . . . . . . . . . . 12 |- ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( -u A ^c 1 )
83		cxp1 23220	. . . . . . . . . . . . 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 2507	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = -u A )
86		rpcxpcl 23225	. . . . . . . . . . . . . . 15 |- ( ( -u A e. RR+ /\ ( 1 / 2 ) e. RR ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
87	39, 2, 86	sylancl 660	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
88	87	rpcnd 11261	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. CC )
89	88	sqvald 12289	. . . . . . . . . . . 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 23228	. . . . . . . . . . . . 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 1232	. . . . . . . . . . . 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 2498	. . . . . . . . . . 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 13352	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) ^ 2 ) = -u A )
94	85, 92, 93	3eqtr4d 2505	. . . . . . . . . 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 11266	. . . . . . . . . . 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 13325	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u A ) e. RR+ )
97	96	rprege0d 11266	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) e. RR /\ 0 <_ ( sqr ` -u A ) ) )
98		sq11 12222	. . . . . . . . . . 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 659	. . . . . . . . . 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 2497	. . . . . . . 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 6288	. . . . . . 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 2499	. . . . . 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 23214	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
105	1, 104	mp3an3 1311	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
106	105	adantr 463	. . . . . 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 5852	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u -u A ) = ( sqr ` A ) )
108	39	rpge0d 11263	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 <_ -u A )
109	28, 108	sqrtnegd 13335	. . . . . . 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 2497	. . . . . 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 2505	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
112	111	ex 432	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
113	81	oveq2i 6281	. . . . . . . . 9 |- ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( A ^c 1 )
114		cxpadd 23228	. . . . . . . . . 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 1314	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
116		cxp1 23220	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c 1 ) = A )
117	116	adantr 463	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c 1 ) = A )
118	113, 115, 117	3eqtr3a 2519	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) = A )
119		cxpcl 23223	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) e. CC )
120	1, 119	mpan2 669	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) e. CC )
121	120	sqvald 12289	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
122	121	adantr 463	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
123	20	adantr 463	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( sqr ` A ) ^ 2 ) = A )
124	118, 122, 123	3eqtr4d 2505	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) )
125		sqeqor 12264	. . . . . . . . 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 659	. . . . . . . 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 482	. . . . . . 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 468	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
129	128	ord 375	. . . . 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 432	. 2 |- ( A e. CC -> ( A =/= 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
133	12, 132	pm2.61dne 2771	1 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483   + caddc 9484   x. cmul 9486   < clt 9617   <_ cle 9618   -ucneg 9797   / cdiv 10202   2c2 10581   RR+crp 11221   ^cexp 12148   sqrcsqrt 13148   expce 13879   picpi 13884   logclog 23108   ^c ccxp 23109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111

This theorem is referenced by:  logsqrt  23253  dvsqrt  23286  resqrtcn  23291  sqrtcn  23292  efiatan  23440  efiatan2  23445  sqrtlim  23500  chpchtlim  23862  dvcnsqrt  30341