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Theorem sqrtneg 14008
Description: The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
Assertion
Ref Expression
sqrtneg |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` -u A ) = ( _i x. ( sqr ` A ) ) )
Proof of Theorem sqrtneg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 recn 10026 . . . . 5 |- ( A e. RR -> A e. CC )
21adantr 481 . . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
32negcld 10379 . . 3 |- ( ( A e. RR /\ 0 <_ A ) -> -u A e. CC )
4 sqrtval 13977 . . 3 |- ( -u A e. CC -> ( sqr ` -u A ) = ( iota_ x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
53, 4syl 17 . 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` -u A ) = ( iota_ x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
6 sqrtneglem 14007 . . 3 |- ( ( 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+ ) )
7 ax-icn 9995 . . . . 5 |- _i e. CC
8 resqrtcl 13994 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. RR )
98recnd 10068 . . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. CC )
10 mulcl 10020 . . . . 5 |- ( ( _i e. CC /\ ( sqr ` A ) e. CC ) -> ( _i x. ( sqr ` A ) ) e. CC )
117, 9, 10sylancr 695 . . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( _i x. ( sqr ` A ) ) e. CC )
12 oveq1 6657 . . . . . . . . 9 |- ( x = ( _i x. ( sqr ` A ) ) -> ( x ^ 2 ) = ( ( _i x. ( sqr ` A ) ) ^ 2 ) )
1312eqeq1d 2624 . . . . . . . 8 |- ( x = ( _i x. ( sqr ` A ) ) -> ( ( x ^ 2 ) = -u A <-> ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A ) )
14 fveq2 6191 . . . . . . . . 9 |- ( x = ( _i x. ( sqr ` A ) ) -> ( Re ` x ) = ( Re ` ( _i x. ( sqr ` A ) ) ) )
1514breq2d 4665 . . . . . . . 8 |- ( x = ( _i x. ( sqr ` A ) ) -> ( 0 <_ ( Re ` x ) <-> 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) ) )
16 oveq2 6658 . . . . . . . . 9 |- ( x = ( _i x. ( sqr ` A ) ) -> ( _i x. x ) = ( _i x. ( _i x. ( sqr ` A ) ) ) )
17 neleq1 2902 . . . . . . . . 9 |- ( ( _i x. x ) = ( _i x. ( _i x. ( sqr ` A ) ) ) -> ( ( _i x. x ) e/ RR+ <-> ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )
1816, 17syl 17 . . . . . . . 8 |- ( x = ( _i x. ( sqr ` A ) ) -> ( ( _i x. x ) e/ RR+ <-> ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )
1913, 15, 183anbi123d 1399 . . . . . . 7 |- ( x = ( _i x. ( sqr ` A ) ) -> ( ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) <-> ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) ) )
2019rspcev 3309 . . . . . 6 |- ( ( ( _i x. ( sqr ` A ) ) e. CC /\ ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) ) -> E. x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
2111, 6, 20syl2anc 693 . . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> E. x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
22 sqrmo 13992 . . . . . 6 |- ( -u A e. CC -> E* x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
233, 22syl 17 . . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> E* x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
24 reu5 3159 . . . . 5 |- ( E! x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) <-> ( E. x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) /\ E* x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
2521, 23, 24sylanbrc 698 . . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> E! x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
2619riota2 6633 . . . 4 |- ( ( ( _i x. ( sqr ` A ) ) e. CC /\ E! x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) -> ( ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) <-> ( iota_ x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( _i x. ( sqr ` A ) ) ) )
2711, 25, 26syl2anc 693 . . 3 |- ( ( 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+ ) <-> ( iota_ x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( _i x. ( sqr ` A ) ) ) )
286, 27mpbid 222 . 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( iota_ x e. CC ( ( x ^ 2 ) = -u A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( _i x. ( sqr ` A ) ) )
295, 28eqtrd 2656 1 |- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` -u A ) = ( _i x. ( sqr ` A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   E.wrex 2913   E!wreu 2914   E*wrmo 2915   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   x. cmul 9941   <_ cle 10075   -ucneg 10267   2c2 11070   RR+crp 11832   ^cexp 12860   Recre 13837   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:  sqrtm1  14016  sqrtnegd  14160
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