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Theorem sqrneglem 11752
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 8796 . . . 4  |-  _i  e.  CC
2 resqrcl 11739 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
3 recn 8827 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  ( sqr `  A )  e.  CC )
42, 3syl 15 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
5 sqmul 11167 . . . 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 11203 . . . . 5  |-  ( _i
^ 2 )  = 
-u 1
87a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i ^ 2 )  =  -u 1
)
9 resqrth 11741 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
108, 9oveq12d 5876 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) )  =  ( -u
1  x.  A ) )
11 recn 8827 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
1211adantr 451 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
1312mulm1d 9231 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u 1  x.  A
)  =  -u A
)
146, 10, 133eqtrd 2319 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  -u A )
15 renegcl 9110 . . . 4  |-  ( ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  RR )
16 0re 8838 . . . . 5  |-  0  e.  RR
17 reim0 11603 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  0 )
18 recn 8827 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  CC )
19 imre 11593 . . . . . . 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 2317 . . . . 5  |-  ( -u ( sqr `  A )  e.  RR  ->  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
22 eqle 8923 . . . . 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 9219 . . . . 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 5529 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( Re `  ( -u _i  x.  -u ( sqr `  A ) ) )  =  ( Re
`  ( _i  x.  ( sqr `  A ) ) ) )
2824, 27breqtrd 4047 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( _i  x.  ( sqr `  A ) ) ) )
29 sqrge0 11743 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( sqr `  A ) )
30 le0neg2 9283 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -u ( sqr `  A )  <_  0
) )
31 lenlt 8901 . . . . . . . . 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 10356 . . . . . 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 9397 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
4241oveq1i 5868 . . . . . . 7  |-  ( ( _i  x.  _i )  x.  ( sqr `  A
) )  =  (
-u 1  x.  ( sqr `  A ) )
43 mulass 8825 . . . . . . . 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 9221 . . . . . . 7  |-  ( ( sqr `  A )  e.  CC  ->  ( -u 1  x.  ( sqr `  A ) )  = 
-u ( sqr `  A
) )
4642, 44, 453eqtr3a 2339 . . . . . 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 2349 . . . 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 2449 . . 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 1623    e. wcel 1684    e/ wnel 2447   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742    < clt 8867    <_ cle 8868   -ucneg 9038   2c2 9795   RR+crp 10354   ^cexp 11104   Recre 11582   Imcim 11583   sqrcsqr 11718
This theorem is referenced by:  sqrneg  11753  sqreu  11844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720
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