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Theorem sqrneg 12001
Description: The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
Assertion
Ref Expression
sqrneg  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  -u A
)  =  ( _i  x.  ( sqr `  A
) ) )

Proof of Theorem sqrneg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 recn 9014 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
32negcld 9331 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -u A  e.  CC )
4 sqrval 11970 . . 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 16 . 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 sqrneglem 12000 . . 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 8983 . . . . 5  |-  _i  e.  CC
8 resqrcl 11987 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
98recnd 9048 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
10 mulcl 9008 . . . . 5  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( _i  x.  ( sqr `  A ) )  e.  CC )
117, 9, 10sylancr 645 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  ( sqr `  A ) )  e.  CC )
12 oveq1 6028 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( x ^ 2 )  =  ( ( _i  x.  ( sqr `  A ) ) ^ 2 ) )
1312eqeq1d 2396 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( (
x ^ 2 )  =  -u A  <->  ( (
_i  x.  ( sqr `  A ) ) ^
2 )  =  -u A ) )
14 fveq2 5669 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( Re `  x )  =  ( Re `  ( _i  x.  ( sqr `  A
) ) ) )
1514breq2d 4166 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( 0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( _i  x.  ( sqr `  A
) ) ) ) )
16 oveq2 6029 . . . . . . . . 9  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( _i  x.  x )  =  ( _i  x.  ( _i  x.  ( sqr `  A
) ) ) )
17 neleq1 2644 . . . . . . . . 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 16 . . . . . . . 8  |-  ( x  =  ( _i  x.  ( sqr `  A ) )  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( _i  x.  ( sqr `  A ) ) )  e/  RR+ )
)
1913, 15, 183anbi123d 1254 . . . . . . 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 2996 . . . . . 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 643 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E. x  e.  CC  ( ( x ^
2 )  =  -u A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
22 sqrmo 11985 . . . . . 6  |-  ( -u A  e.  CC  ->  E* x  e.  CC ( ( x ^ 2 )  =  -u A  /\  0  <_  ( Re
`  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
233, 22syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E* x  e.  CC ( ( x ^
2 )  =  -u A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
24 reu5 2865 . . . . 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 646 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E! x  e.  CC  ( ( x ^
2 )  =  -u A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
2619riota2 6509 . . . 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 643 . . 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 202 . 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 2420 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  -u A
)  =  ( _i  x.  ( sqr `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e/ wnel 2552   E.wrex 2651   E!wreu 2652   E*wrmo 2653   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   iota_crio 6479   CCcc 8922   RRcr 8923   0cc0 8924   _ici 8926    x. cmul 8929    <_ cle 9055   -ucneg 9225   2c2 9982   RR+crp 10545   ^cexp 11310   Recre 11830   sqrcsqr 11966
This theorem is referenced by:  sqrm1  12009  sqrnegd  12152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968
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