Theorem cxpsqrt 24449

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 11247	. . . . . 6 |- ( 1 / 2 ) e. CC
2		halfre 11246	. . . . . . 7 |- ( 1 / 2 ) e. RR
3		halfgt0 11248	. . . . . . 7 |- 0 < ( 1 / 2 )
4	2, 3	gt0ne0ii 10564	. . . . . 6 |- ( 1 / 2 ) =/= 0
5		0cxp 24412	. . . . . 6 |- ( ( ( 1 / 2 ) e. CC /\ ( 1 / 2 ) =/= 0 ) -> ( 0 ^c ( 1 / 2 ) ) = 0 )
6	1, 4, 5	mp2an 708	. . . . 5 |- ( 0 ^c ( 1 / 2 ) ) = 0
7		sqrt0 13982	. . . . 5 |- ( sqr ` 0 ) = 0
8	6, 7	eqtr4i 2647	. . . 4 |- ( 0 ^c ( 1 / 2 ) ) = ( sqr ` 0 )
9		oveq1 6657	. . . 4 |- ( A = 0 -> ( A ^c ( 1 / 2 ) ) = ( 0 ^c ( 1 / 2 ) ) )
10		fveq2 6191	. . . 4 |- ( A = 0 -> ( sqr ` A ) = ( sqr ` 0 ) )
11	8, 9, 10	3eqtr4a 2682	. . 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 9995	. . . . . . . . . . . . . . . . 17 |- _i e. CC
14		sqrtcl 14101	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( sqr ` A ) e. CC )
15	14	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` A ) e. CC )
16		sqmul 12926	. . . . . . . . . . . . . . . . 17 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
17	13, 15, 16	sylancr 695	. . . . . . . . . . . . . . . 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 12965	. . . . . . . . . . . . . . . . . 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 14104	. . . . . . . . . . . . . . . . . 18 |- ( A e. CC -> ( ( sqr ` A ) ^ 2 ) = A )
21	20	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` A ) ^ 2 ) = A )
22	19, 21	oveq12d 6668	. . . . . . . . . . . . . . . 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 10471	. . . . . . . . . . . . . . . . 17 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
24	23	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u 1 x. A ) = -u A )
25	17, 22, 24	3eqtrd 2660	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A )
26		cxpsqrtlem 24448	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( _i x. ( sqr ` A ) ) e. RR )
27	26	resqcld 13035	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) e. RR )
28	25, 27	eqeltrrd 2702	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR )
29		negeq0 10335	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
30	29	necon3bid 2838	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) )
31	30	biimpa 501	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 )
32	31	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A =/= 0 )
33	25, 32	eqnetrd 2861	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) =/= 0 )
34		sq0i 12956	. . . . . . . . . . . . . . . . . 18 |- ( ( _i x. ( sqr ` A ) ) = 0 -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = 0 )
35	34	necon3i 2826	. . . . . . . . . . . . . . . . 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 13037	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < ( ( _i x. ( sqr ` A ) ) ^ 2 ) )
38	37, 25	breqtrd 4679	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 < -u A )
39	28, 38	elrpd 11869	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u A e. RR+ )
40		logneg 24334	. . . . . . . . . . . . 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 10331	. . . . . . . . . . . . . 14 |- ( A e. CC -> -u -u A = A )
43	42	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> -u -u A = A )
44	43	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u -u A ) = ( log ` A ) )
45	39	relogcld 24369	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. RR )
46	45	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` -u A ) e. CC )
47		picn 24211	. . . . . . . . . . . . . 14 |- pi e. CC
48	13, 47	mulcli 10045	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
49		addcom 10222	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 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 2664	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( log ` A ) = ( ( _i x. pi ) + ( log ` -u A ) ) )
52	51	oveq2d 6666	. . . . . . . . . 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 10025	. . . . . . . . . . . 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 1414	. . . . . . . . . . 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 2656	. . . . . . . . 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 11091	. . . . . . . . . . . 12 |- 2 e. CC
58		2ne0 11113	. . . . . . . . . . . 12 |- 2 =/= 0
59		divrec2 10702	. . . . . . . . . . . 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 1424	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( ( 1 / 2 ) x. ( _i x. pi ) )
61	13, 47, 57, 58	divassi 10781	. . . . . . . . . . 11 |- ( ( _i x. pi ) / 2 ) = ( _i x. ( pi / 2 ) )
62	60, 61	eqtr3i 2646	. . . . . . . . . 10 |- ( ( 1 / 2 ) x. ( _i x. pi ) ) = ( _i x. ( pi / 2 ) )
63	62	oveq1i 6660	. . . . . . . . 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 2672	. . . . . . . 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 6195	. . . . . . 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 10767	. . . . . . . . 9 |- ( pi / 2 ) e. CC
67	13, 66	mulcli 10045	. . . . . . . 8 |- ( _i x. ( pi / 2 ) ) e. CC
68		mulcl 10020	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` -u A ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
69	1, 46, 68	sylancr 695	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( 1 / 2 ) x. ( log ` -u A ) ) e. CC )
70		efadd 14824	. . . . . . . 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 695	. . . . . . 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 24223	. . . . . . . . 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 10281	. . . . . . . . . . 11 |- ( A e. CC -> -u A e. CC )
75	74	ad2antrr 762	. . . . . . . . . 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 24411	. . . . . . . . . 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 1326	. . . . . . . . 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 9994	. . . . . . . . . . . . . 14 |- 1 e. CC
80		2halves 11260	. . . . . . . . . . . . . 14 |- ( 1 e. CC -> ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 )
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
82	81	oveq2i 6661	. . . . . . . . . . . 12 |- ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( -u A ^c 1 )
83		cxp1 24417	. . . . . . . . . . . . 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 2668	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = -u A )
86		rpcxpcl 24422	. . . . . . . . . . . . . . 15 |- ( ( -u A e. RR+ /\ ( 1 / 2 ) e. RR ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
87	39, 2, 86	sylancl 694	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. RR+ )
88	87	rpcnd 11874	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( -u A ^c ( 1 / 2 ) ) e. CC )
89	88	sqvald 13005	. . . . . . . . . . . 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 24425	. . . . . . . . . . . . 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 1332	. . . . . . . . . . . 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 2659	. . . . . . . . . . 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 14178	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) ^ 2 ) = -u A )
94	85, 92, 93	3eqtr4d 2666	. . . . . . . . . 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 11879	. . . . . . . . . . 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 14150	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u A ) e. RR+ )
97	96	rprege0d 11879	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( ( sqr ` -u A ) e. RR /\ 0 <_ ( sqr ` -u A ) ) )
98		sq11 12936	. . . . . . . . . . 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 693	. . . . . . . . . 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 222	. . . . . . . . 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 2658	. . . . . . . 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 6668	. . . . . . 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 2660	. . . . . 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 24411	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
105	1, 104	mp3an3 1413	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` A ) ) ) )
106	105	adantr 481	. . . . . 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 6195	. . . . . . 7 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( sqr ` -u -u A ) = ( sqr ` A ) )
108	39	rpge0d 11876	. . . . . . . 8 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> 0 <_ -u A )
109	28, 108	sqrtnegd 14160	. . . . . . 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 2658	. . . . . 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 2666	. . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
112	111	ex 450	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
113	81	oveq2i 6661	. . . . . . . . 9 |- ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( A ^c 1 )
114		cxpadd 24425	. . . . . . . . . 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 1416	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
116		cxp1 24417	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c 1 ) = A )
117	116	adantr 481	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c 1 ) = A )
118	113, 115, 117	3eqtr3a 2680	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) = A )
119		cxpcl 24420	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( 1 / 2 ) e. CC ) -> ( A ^c ( 1 / 2 ) ) e. CC )
120	1, 119	mpan2 707	. . . . . . . . . 10 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) e. CC )
121	120	sqvald 13005	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
122	121	adantr 481	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( A ^c ( 1 / 2 ) ) x. ( A ^c ( 1 / 2 ) ) ) )
123	20	adantr 481	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( sqr ` A ) ^ 2 ) = A )
124	118, 122, 123	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) ^ 2 ) = ( ( sqr ` A ) ^ 2 ) )
125		sqeqor 12978	. . . . . . . . 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 693	. . . . . . . 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 501	. . . . . . 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 487	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) \/ ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
129	128	ord 392	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) ) )
130	129	con1d 139	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( A ^c ( 1 / 2 ) ) = -u ( sqr ` A ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
131	112, 130	pm2.61d 170	. . 3 |- ( ( A e. CC /\ A =/= 0 ) -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
132	131	ex 450	. 2 |- ( A e. CC -> ( A =/= 0 -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) ) )
133	12, 132	pm2.61dne 2880	1 |- ( A e. CC -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   2c2 11070   RR+crp 11832   ^cexp 12860   sqrcsqrt 13973   expce 14792   picpi 14797   logclog 24301   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  logsqrt  24450  dvsqrt  24483  dvcnsqrt  24485  resqrtcn  24490  sqrtcn  24491  efiatan  24639  efiatan2  24644  sqrtlim  24699  chpchtlim  25168  logdivsqrle  30728