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Theorem sqrneglem 12000
Description: The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
Assertion
Ref Expression
sqrneglem  |-  ( ( 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 sqrneglem
StepHypRef Expression
1 ax-icn 8983 . . . 4  |-  _i  e.  CC
2 resqrcl 11987 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
3 recn 9014 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  ( sqr `  A )  e.  CC )
42, 3syl 16 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
5 sqmul 11373 . . . 4  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
61, 4, 5sylancr 645 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
7 i2 11409 . . . . 5  |-  ( _i
^ 2 )  = 
-u 1
87a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i ^ 2 )  =  -u 1
)
9 resqrth 11989 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
108, 9oveq12d 6039 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) )  =  ( -u
1  x.  A ) )
11 recn 9014 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
1211adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
1312mulm1d 9418 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u 1  x.  A
)  =  -u A
)
146, 10, 133eqtrd 2424 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  -u A )
15 renegcl 9297 . . . 4  |-  ( ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  RR )
16 0re 9025 . . . . 5  |-  0  e.  RR
17 reim0 11851 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  0 )
18 recn 9014 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  CC )
19 imre 11841 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  CC  ->  (
Im `  -u ( sqr `  A ) )  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
2018, 19syl 16 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
2117, 20eqtr3d 2422 . . . . 5  |-  ( -u ( sqr `  A )  e.  RR  ->  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
22 eqle 9110 . . . . 5  |-  ( ( 0  e.  RR  /\  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )  ->  0  <_  (
Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
2316, 21, 22sylancr 645 . . . 4  |-  ( -u ( sqr `  A )  e.  RR  ->  0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
242, 15, 233syl 19 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
25 mul2neg 9406 . . . . 5  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
261, 4, 25sylancr 645 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
2726fveq2d 5673 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( Re `  ( -u _i  x.  -u ( sqr `  A ) ) )  =  ( Re
`  ( _i  x.  ( sqr `  A ) ) ) )
2824, 27breqtrd 4178 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( _i  x.  ( sqr `  A ) ) ) )
29 ixi 9584 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
3029oveq1i 6031 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( sqr `  A
) )  =  (
-u 1  x.  ( sqr `  A ) )
31 mulass 9012 . . . . . . 7  |-  ( ( _i  e.  CC  /\  _i  e.  CC  /\  ( sqr `  A )  e.  CC )  ->  (
( _i  x.  _i )  x.  ( sqr `  A ) )  =  ( _i  x.  (
_i  x.  ( sqr `  A ) ) ) )
321, 1, 31mp3an12 1269 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  (
( _i  x.  _i )  x.  ( sqr `  A ) )  =  ( _i  x.  (
_i  x.  ( sqr `  A ) ) ) )
33 mulm1 9408 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  ( -u 1  x.  ( sqr `  A ) )  = 
-u ( sqr `  A
) )
3430, 32, 333eqtr3a 2444 . . . . 5  |-  ( ( sqr `  A )  e.  CC  ->  (
_i  x.  ( _i  x.  ( sqr `  A
) ) )  = 
-u ( sqr `  A
) )
354, 34syl 16 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  (
_i  x.  ( sqr `  A ) ) )  =  -u ( sqr `  A
) )
36 sqrge0 11991 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( sqr `  A ) )
37 le0neg2 9470 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -u ( sqr `  A )  <_  0
) )
38 lenlt 9088 . . . . . . . . 9  |-  ( (
-u ( sqr `  A
)  e.  RR  /\  0  e.  RR )  ->  ( -u ( sqr `  A )  <_  0  <->  -.  0  <  -u ( sqr `  A ) ) )
3915, 16, 38sylancl 644 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  ( -u ( sqr `  A
)  <_  0  <->  -.  0  <  -u ( sqr `  A
) ) )
4037, 39bitrd 245 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -.  0  <  -u ( sqr `  A
) ) )
412, 40syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0  <_  ( sqr `  A )  <->  -.  0  <  -u ( sqr `  A
) ) )
4236, 41mpbid 202 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  0  <  -u ( sqr `  A ) )
432, 15syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -u ( sqr `  A
)  e.  RR )
4443biantrurd 495 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0  <  -u ( sqr `  A )  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) ) )
45 elrp 10547 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR+  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) )
4644, 45syl6rbbr 256 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u ( sqr `  A
)  e.  RR+  <->  0  <  -u ( sqr `  A
) ) )
4742, 46mtbird 293 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  -u ( sqr `  A
)  e.  RR+ )
4835, 47eqneltrd 2481 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
49 df-nel 2554 . . 3  |-  ( ( _i  x.  ( _i  x.  ( sqr `  A
) ) )  e/  RR+  <->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
5048, 49sylibr 204 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e/  RR+ )
5114, 28, 503jca 1134 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+ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e/ wnel 2552   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925   _ici 8926    x. cmul 8929    < clt 9054    <_ cle 9055   -ucneg 9225   2c2 9982   RR+crp 10545   ^cexp 11310   Recre 11830   Imcim 11831   sqrcsqr 11966
This theorem is referenced by:  sqrneg  12001  sqreu  12092
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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