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Theorem sqrneglem 12064
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 9041 . . . 4  |-  _i  e.  CC
2 resqrcl 12051 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
3 recn 9072 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  ( sqr `  A )  e.  CC )
42, 3syl 16 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
5 sqmul 11437 . . . 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 11473 . . . . 5  |-  ( _i
^ 2 )  = 
-u 1
87a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i ^ 2 )  =  -u 1
)
9 resqrth 12053 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
108, 9oveq12d 6091 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) )  =  ( -u
1  x.  A ) )
11 recn 9072 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
1211adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
1312mulm1d 9477 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u 1  x.  A
)  =  -u A
)
146, 10, 133eqtrd 2471 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  -u A )
15 renegcl 9356 . . . 4  |-  ( ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  RR )
16 0re 9083 . . . . 5  |-  0  e.  RR
17 reim0 11915 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  0 )
18 recn 9072 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  CC )
19 imre 11905 . . . . . . 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 2469 . . . . 5  |-  ( -u ( sqr `  A )  e.  RR  ->  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
22 eqle 9168 . . . . 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 9465 . . . . 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 5724 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( Re `  ( -u _i  x.  -u ( sqr `  A ) ) )  =  ( Re
`  ( _i  x.  ( sqr `  A ) ) ) )
2824, 27breqtrd 4228 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( _i  x.  ( sqr `  A ) ) ) )
29 ixi 9643 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
3029oveq1i 6083 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( sqr `  A
) )  =  (
-u 1  x.  ( sqr `  A ) )
31 mulass 9070 . . . . . . 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 9467 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  ( -u 1  x.  ( sqr `  A ) )  = 
-u ( sqr `  A
) )
3430, 32, 333eqtr3a 2491 . . . . 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 12055 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( sqr `  A ) )
37 le0neg2 9529 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -u ( sqr `  A )  <_  0
) )
38 lenlt 9146 . . . . . . . . 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 10606 . . . . . 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 2528 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
49 df-nel 2601 . . 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 1652    e. wcel 1725    e/ wnel 2599   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983   _ici 8984    x. cmul 8987    < clt 9112    <_ cle 9113   -ucneg 9284   2c2 10041   RR+crp 10604   ^cexp 11374   Recre 11894   Imcim 11895   sqrcsqr 12030
This theorem is referenced by:  sqrneg  12065  sqreu  12156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032
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