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Theorem sqrneglem 11768
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 8812 . . . 4  |-  _i  e.  CC
2 resqrcl 11755 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
3 recn 8843 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  ( sqr `  A )  e.  CC )
42, 3syl 15 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
5 sqmul 11183 . . . 4  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
61, 4, 5sylancr 644 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
7 i2 11219 . . . . 5  |-  ( _i
^ 2 )  = 
-u 1
87a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i ^ 2 )  =  -u 1
)
9 resqrth 11757 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
108, 9oveq12d 5892 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) )  =  ( -u
1  x.  A ) )
11 recn 8843 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
1211adantr 451 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
1312mulm1d 9247 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u 1  x.  A
)  =  -u A
)
146, 10, 133eqtrd 2332 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  -u A )
15 renegcl 9126 . . . 4  |-  ( ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  RR )
16 0re 8854 . . . . 5  |-  0  e.  RR
17 reim0 11619 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  0 )
18 recn 8843 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  CC )
19 imre 11609 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  CC  ->  (
Im `  -u ( sqr `  A ) )  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
2018, 19syl 15 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
2117, 20eqtr3d 2330 . . . . 5  |-  ( -u ( sqr `  A )  e.  RR  ->  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
22 eqle 8939 . . . . 5  |-  ( ( 0  e.  RR  /\  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )  ->  0  <_  (
Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
2316, 21, 22sylancr 644 . . . 4  |-  ( -u ( sqr `  A )  e.  RR  ->  0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
242, 15, 233syl 18 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
25 mul2neg 9235 . . . . 5  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
261, 4, 25sylancr 644 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
2726fveq2d 5545 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( Re `  ( -u _i  x.  -u ( sqr `  A ) ) )  =  ( Re
`  ( _i  x.  ( sqr `  A ) ) ) )
2824, 27breqtrd 4063 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( _i  x.  ( sqr `  A ) ) ) )
29 sqrge0 11759 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( sqr `  A ) )
30 le0neg2 9299 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -u ( sqr `  A )  <_  0
) )
31 lenlt 8917 . . . . . . . . 9  |-  ( (
-u ( sqr `  A
)  e.  RR  /\  0  e.  RR )  ->  ( -u ( sqr `  A )  <_  0  <->  -.  0  <  -u ( sqr `  A ) ) )
3215, 16, 31sylancl 643 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  ( -u ( sqr `  A
)  <_  0  <->  -.  0  <  -u ( sqr `  A
) ) )
3330, 32bitrd 244 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -.  0  <  -u ( sqr `  A
) ) )
342, 33syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0  <_  ( sqr `  A )  <->  -.  0  <  -u ( sqr `  A
) ) )
3529, 34mpbid 201 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  0  <  -u ( sqr `  A ) )
362, 15syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -u ( sqr `  A
)  e.  RR )
3736biantrurd 494 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0  <  -u ( sqr `  A )  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) ) )
38 elrp 10372 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR+  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) )
3937, 38syl6rbbr 255 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u ( sqr `  A
)  e.  RR+  <->  0  <  -u ( sqr `  A
) ) )
4035, 39mtbird 292 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  -u ( sqr `  A
)  e.  RR+ )
41 ixi 9413 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
4241oveq1i 5884 . . . . . . 7  |-  ( ( _i  x.  _i )  x.  ( sqr `  A
) )  =  (
-u 1  x.  ( sqr `  A ) )
43 mulass 8841 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  _i  e.  CC  /\  ( sqr `  A )  e.  CC )  ->  (
( _i  x.  _i )  x.  ( sqr `  A ) )  =  ( _i  x.  (
_i  x.  ( sqr `  A ) ) ) )
441, 1, 43mp3an12 1267 . . . . . . 7  |-  ( ( sqr `  A )  e.  CC  ->  (
( _i  x.  _i )  x.  ( sqr `  A ) )  =  ( _i  x.  (
_i  x.  ( sqr `  A ) ) ) )
45 mulm1 9237 . . . . . . 7  |-  ( ( sqr `  A )  e.  CC  ->  ( -u 1  x.  ( sqr `  A ) )  = 
-u ( sqr `  A
) )
4642, 44, 453eqtr3a 2352 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  (
_i  x.  ( _i  x.  ( sqr `  A
) ) )  = 
-u ( sqr `  A
) )
474, 46syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  (
_i  x.  ( sqr `  A ) ) )  =  -u ( sqr `  A
) )
4847eleq1d 2362 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( _i  x.  ( sqr `  A ) ) )  e.  RR+  <->  -u ( sqr `  A )  e.  RR+ ) )
4940, 48mtbird 292 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
50 df-nel 2462 . . 3  |-  ( ( _i  x.  ( _i  x.  ( sqr `  A
) ) )  e/  RR+  <->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
5149, 50sylibr 203 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e/  RR+ )
5214, 28, 513jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e/ wnel 2460   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    x. cmul 8758    < clt 8883    <_ cle 8884   -ucneg 9054   2c2 9811   RR+crp 10370   ^cexp 11120   Recre 11598   Imcim 11599   sqrcsqr 11734
This theorem is referenced by:  sqrneg  11769  sqreu  11860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736
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