Theorem sqrtneglem 14007

Description: The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Assertion

Ref	Expression
sqrtneglem	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )

Proof of Theorem sqrtneglem

Step	Hyp	Ref	Expression
1		ax-icn 9995	. . . 4 |- _i e. CC
2		resqrtcl 13994	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. RR )
3		recn 10026	. . . . 5 |- ( ( sqr ` A ) e. RR -> ( sqr ` A ) e. CC )
4	2, 3	syl 17	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. CC )
5		sqmul 12926	. . . 4 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
6	1, 4, 5	sylancr 695	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) )
7		i2 12965	. . . . 5 |- ( _i ^ 2 ) = -u 1
8	7	a1i 11	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( _i ^ 2 ) = -u 1 )
9		resqrtth 13996	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) ^ 2 ) = A )
10	8, 9	oveq12d 6668	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( _i ^ 2 ) x. ( ( sqr ` A ) ^ 2 ) ) = ( -u 1 x. A ) )
11		recn 10026	. . . . 5 |- ( A e. RR -> A e. CC )
12	11	adantr 481	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
13	12	mulm1d 10482	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( -u 1 x. A ) = -u A )
14	6, 10, 13	3eqtrd 2660	. 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A )
15		renegcl 10344	. . . 4 |- ( ( sqr ` A ) e. RR -> -u ( sqr ` A ) e. RR )
16		0re 10040	. . . . 5 |- 0 e. RR
17		reim0 13858	. . . . . 6 |- ( -u ( sqr ` A ) e. RR -> ( Im ` -u ( sqr ` A ) ) = 0 )
18		recn 10026	. . . . . . 7 |- ( -u ( sqr ` A ) e. RR -> -u ( sqr ` A ) e. CC )
19		imre 13848	. . . . . . 7 |- ( -u ( sqr ` A ) e. CC -> ( Im ` -u ( sqr ` A ) ) = ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
20	18, 19	syl 17	. . . . . 6 |- ( -u ( sqr ` A ) e. RR -> ( Im ` -u ( sqr ` A ) ) = ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
21	17, 20	eqtr3d 2658	. . . . 5 |- ( -u ( sqr ` A ) e. RR -> 0 = ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
22		eqle 10139	. . . . 5 |- ( ( 0 e. RR /\ 0 = ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) ) -> 0 <_ ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
23	16, 21, 22	sylancr 695	. . . 4 |- ( -u ( sqr ` A ) e. RR -> 0 <_ ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
24	2, 15, 23	3syl 18	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) )
25		mul2neg 10469	. . . . 5 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( -u _i x. -u ( sqr ` A ) ) = ( _i x. ( sqr ` A ) ) )
26	1, 4, 25	sylancr 695	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( -u _i x. -u ( sqr ` A ) ) = ( _i x. ( sqr ` A ) ) )
27	26	fveq2d 6195	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( Re ` ( -u _i x. -u ( sqr ` A ) ) ) = ( Re ` ( _i x. ( sqr ` A ) ) ) )
28	24, 27	breqtrd 4679	. 2 |- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) )
29		ixi 10656	. . . . . . 7 |- ( _i x. _i ) = -u 1
30	29	oveq1i 6660	. . . . . 6 |- ( ( _i x. _i ) x. ( sqr ` A ) ) = ( -u 1 x. ( sqr ` A ) )
31		mulass 10024	. . . . . . 7 |- ( ( _i e. CC /\ _i e. CC /\ ( sqr ` A ) e. CC ) -> ( ( _i x. _i ) x. ( sqr ` A ) ) = ( _i x. ( _i x. ( sqr ` A ) ) ) )
32	1, 1, 31	mp3an12 1414	. . . . . 6 |- ( ( sqr ` A ) e. CC -> ( ( _i x. _i ) x. ( sqr ` A ) ) = ( _i x. ( _i x. ( sqr ` A ) ) ) )
33		mulm1 10471	. . . . . 6 |- ( ( sqr ` A ) e. CC -> ( -u 1 x. ( sqr ` A ) ) = -u ( sqr ` A ) )
34	30, 32, 33	3eqtr3a 2680	. . . . 5 |- ( ( sqr ` A ) e. CC -> ( _i x. ( _i x. ( sqr ` A ) ) ) = -u ( sqr ` A ) )
35	4, 34	syl 17	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( _i x. ( _i x. ( sqr ` A ) ) ) = -u ( sqr ` A ) )
36		sqrtge0 13998	. . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqr ` A ) )
37		le0neg2 10537	. . . . . . . 8 |- ( ( sqr ` A ) e. RR -> ( 0 <_ ( sqr ` A ) <-> -u ( sqr ` A ) <_ 0 ) )
38		lenlt 10116	. . . . . . . . 9 |- ( ( -u ( sqr ` A ) e. RR /\ 0 e. RR ) -> ( -u ( sqr ` A ) <_ 0 <-> -. 0 < -u ( sqr ` A ) ) )
39	15, 16, 38	sylancl 694	. . . . . . . 8 |- ( ( sqr ` A ) e. RR -> ( -u ( sqr ` A ) <_ 0 <-> -. 0 < -u ( sqr ` A ) ) )
40	37, 39	bitrd 268	. . . . . . 7 |- ( ( sqr ` A ) e. RR -> ( 0 <_ ( sqr ` A ) <-> -. 0 < -u ( sqr ` A ) ) )
41	2, 40	syl 17	. . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( 0 <_ ( sqr ` A ) <-> -. 0 < -u ( sqr ` A ) ) )
42	36, 41	mpbid 222	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> -. 0 < -u ( sqr ` A ) )
43	2, 15	syl 17	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A ) -> -u ( sqr ` A ) e. RR )
44	43	biantrurd 529	. . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( 0 < -u ( sqr ` A ) <-> ( -u ( sqr ` A ) e. RR /\ 0 < -u ( sqr ` A ) ) ) )
45		elrp 11834	. . . . . 6 |- ( -u ( sqr ` A ) e. RR+ <-> ( -u ( sqr ` A ) e. RR /\ 0 < -u ( sqr ` A ) ) )
46	44, 45	syl6rbbr 279	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> ( -u ( sqr ` A ) e. RR+ <-> 0 < -u ( sqr ` A ) ) )
47	42, 46	mtbird 315	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> -. -u ( sqr ` A ) e. RR+ )
48	35, 47	eqneltrd 2720	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> -. ( _i x. ( _i x. ( sqr ` A ) ) ) e. RR+ )
49		df-nel 2898	. . 3 |- ( ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ <-> -. ( _i x. ( _i x. ( sqr ` A ) ) ) e. RR+ )
50	48, 49	sylibr 224	. 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ )
51	14, 28, 50	3jca 1242	1 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   2c2 11070   RR+crp 11832   ^cexp 12860   Recre 13837   Imcim 13838   sqrcsqrt 13973

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975

This theorem is referenced by:  sqrtneg  14008  sqreu  14100