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Theorem sqrneglem 12869
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 9447 . . . 4  |-  _i  e.  CC
2 resqrcl 12856 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
3 recn 9478 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  ( sqr `  A )  e.  CC )
42, 3syl 16 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
5 sqmul 12041 . . . 4  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
61, 4, 5sylancr 663 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( ( sqr `  A
) ^ 2 ) ) )
7 i2 12078 . . . . 5  |-  ( _i
^ 2 )  = 
-u 1
87a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i ^ 2 )  =  -u 1
)
9 resqrth 12858 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
108, 9oveq12d 6213 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i ^
2 )  x.  (
( sqr `  A
) ^ 2 ) )  =  ( -u
1  x.  A ) )
11 recn 9478 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
1211adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
1312mulm1d 9902 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u 1  x.  A
)  =  -u A
)
146, 10, 133eqtrd 2497 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( _i  x.  ( sqr `  A ) ) ^ 2 )  =  -u A )
15 renegcl 9778 . . . 4  |-  ( ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  RR )
16 0re 9492 . . . . 5  |-  0  e.  RR
17 reim0 12720 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR  ->  (
Im `  -u ( sqr `  A ) )  =  0 )
18 recn 9478 . . . . . . 7  |-  ( -u ( sqr `  A )  e.  RR  ->  -u ( sqr `  A )  e.  CC )
19 imre 12710 . . . . . . 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 2495 . . . . 5  |-  ( -u ( sqr `  A )  e.  RR  ->  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
22 eqle 9583 . . . . 5  |-  ( ( 0  e.  RR  /\  0  =  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )  ->  0  <_  (
Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
2316, 21, 22sylancr 663 . . . 4  |-  ( -u ( sqr `  A )  e.  RR  ->  0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A ) ) ) )
242, 15, 233syl 20 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( -u _i  x.  -u ( sqr `  A
) ) ) )
25 mul2neg 9890 . . . . 5  |-  ( ( _i  e.  CC  /\  ( sqr `  A )  e.  CC )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
261, 4, 25sylancr 663 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u _i  x.  -u ( sqr `  A ) )  =  ( _i  x.  ( sqr `  A ) ) )
2726fveq2d 5798 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( Re `  ( -u _i  x.  -u ( sqr `  A ) ) )  =  ( Re
`  ( _i  x.  ( sqr `  A ) ) ) )
2824, 27breqtrd 4419 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( Re `  ( _i  x.  ( sqr `  A ) ) ) )
29 ixi 10071 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
3029oveq1i 6205 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( sqr `  A
) )  =  (
-u 1  x.  ( sqr `  A ) )
31 mulass 9476 . . . . . . 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 1305 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  (
( _i  x.  _i )  x.  ( sqr `  A ) )  =  ( _i  x.  (
_i  x.  ( sqr `  A ) ) ) )
33 mulm1 9892 . . . . . 6  |-  ( ( sqr `  A )  e.  CC  ->  ( -u 1  x.  ( sqr `  A ) )  = 
-u ( sqr `  A
) )
3430, 32, 333eqtr3a 2517 . . . . 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 12860 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  ( sqr `  A ) )
37 le0neg2 9954 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  (
0  <_  ( sqr `  A )  <->  -u ( sqr `  A )  <_  0
) )
38 lenlt 9559 . . . . . . . . 9  |-  ( (
-u ( sqr `  A
)  e.  RR  /\  0  e.  RR )  ->  ( -u ( sqr `  A )  <_  0  <->  -.  0  <  -u ( sqr `  A ) ) )
3915, 16, 38sylancl 662 . . . . . . . 8  |-  ( ( sqr `  A )  e.  RR  ->  ( -u ( sqr `  A
)  <_  0  <->  -.  0  <  -u ( sqr `  A
) ) )
4037, 39bitrd 253 . . . . . . 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 210 . . . . 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 508 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0  <  -u ( sqr `  A )  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) ) )
45 elrp 11099 . . . . . 6  |-  ( -u ( sqr `  A )  e.  RR+  <->  ( -u ( sqr `  A )  e.  RR  /\  0  <  -u ( sqr `  A
) ) )
4644, 45syl6rbbr 264 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( -u ( sqr `  A
)  e.  RR+  <->  0  <  -u ( sqr `  A
) ) )
4742, 46mtbird 301 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  -u ( sqr `  A
)  e.  RR+ )
4835, 47eqneltrd 2561 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
49 df-nel 2648 . . 3  |-  ( ( _i  x.  ( _i  x.  ( sqr `  A
) ) )  e/  RR+  <->  -.  ( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e.  RR+ )
5048, 49sylibr 212 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( _i  x.  (
_i  x.  ( sqr `  A ) ) )  e/  RR+ )
5114, 28, 503jca 1168 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    e/ wnel 2646   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389   _ici 9390    x. cmul 9393    < clt 9524    <_ cle 9525   -ucneg 9702   2c2 10477   RR+crp 11097   ^cexp 11977   Recre 12699   Imcim 12700   sqrcsqr 12835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837
This theorem is referenced by:  sqrneg  12870  sqreu  12961
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