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Theorem sqrthlem 11862
Description: Lemma for sqrth 11864. (Contributed by Mario Carneiro, 10-Jul-2013.)
Assertion
Ref Expression
sqrthlem  |-  ( A  e.  CC  ->  (
( ( sqr `  A
) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)

Proof of Theorem sqrthlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sqrval 11738 . . 3  |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
21eqcomd 2301 . 2  |-  ( A  e.  CC  ->  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )  =  ( sqr `  A
) )
3 sqrcl 11861 . . 3  |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
4 sqreu 11860 . . 3  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
5 oveq1 5881 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( x ^ 2 )  =  ( ( sqr `  A
) ^ 2 ) )
65eqeq1d 2304 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( (
x ^ 2 )  =  A  <->  ( ( sqr `  A ) ^
2 )  =  A ) )
7 fveq2 5541 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( Re `  x )  =  ( Re `  ( sqr `  A ) ) )
87breq2d 4051 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( 0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( sqr `  A ) ) ) )
9 oveq2 5882 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( _i  x.  x )  =  ( _i  x.  ( sqr `  A ) ) )
10 neleq1 2550 . . . . . 6  |-  ( ( _i  x.  x )  =  ( _i  x.  ( sqr `  A ) )  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)
119, 10syl 15 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)
126, 8, 113anbi123d 1252 . . . 4  |-  ( x  =  ( sqr `  A
)  ->  ( (
( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ )  <->  ( (
( sqr `  A
) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
) )
1312riota2 6343 . . 3  |-  ( ( ( sqr `  A
)  e.  CC  /\  E! x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )  ->  ( ( ( ( sqr `  A
) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) )  e/  RR+ )  <->  (
iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )  =  ( sqr `  A
) ) )
143, 4, 13syl2anc 642 . 2  |-  ( A  e.  CC  ->  (
( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A
) )  e/  RR+ )  <->  (
iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )  =  ( sqr `  A
) ) )
152, 14mpbird 223 1  |-  ( A  e.  CC  ->  (
( ( sqr `  A
) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    e/ wnel 2460   E!wreu 2558   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   iota_crio 6313   CCcc 8751   0cc0 8753   _ici 8755    x. cmul 8758    <_ cle 8884   2c2 9811   RR+crp 10370   ^cexp 11120   Recre 11598   sqrcsqr 11734
This theorem is referenced by:  sqrth  11864  sqrrege0  11865  eqsqrd  11867
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  df-abs 11737
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