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Theorem resqrthlem 12019
Description: Lemma for resqrth 12020. (Contributed by Mario Carneiro, 9-Jul-2013.)
Assertion
Ref Expression
resqrthlem  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A
) )  e/  RR+ )
)

Proof of Theorem resqrthlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 recn 9040 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 sqrval 12001 . . . . 5  |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) ) )
32eqcomd 2413 . . . 4  |-  ( A  e.  CC  ->  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )  =  ( sqr `  A
) )
41, 3syl 16 . . 3  |-  ( A  e.  RR  ->  ( iota_ x  e.  CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )  =  ( sqr `  A
) )
54adantr 452 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( iota_ x  e.  CC ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )  =  ( sqr `  A
) )
6 resqrcl 12018 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
76recnd 9074 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  CC )
8 resqreu 12017 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E! x  e.  CC  ( ( x ^
2 )  =  A  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
9 oveq1 6051 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( x ^ 2 )  =  ( ( sqr `  A
) ^ 2 ) )
109eqeq1d 2416 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( (
x ^ 2 )  =  A  <->  ( ( sqr `  A ) ^
2 )  =  A ) )
11 fveq2 5691 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( Re `  x )  =  ( Re `  ( sqr `  A ) ) )
1211breq2d 4188 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( 0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( sqr `  A ) ) ) )
13 oveq2 6052 . . . . . 6  |-  ( x  =  ( sqr `  A
)  ->  ( _i  x.  x )  =  ( _i  x.  ( sqr `  A ) ) )
14 neleq1 2664 . . . . . 6  |-  ( ( _i  x.  x )  =  ( _i  x.  ( sqr `  A ) )  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)
1513, 14syl 16 . . . . 5  |-  ( x  =  ( sqr `  A
)  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
)
1610, 12, 153anbi123d 1254 . . . 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+ )
) )
1716riota2 6535 . . 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
) ) )
187, 8, 17syl2anc 643 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( ( ( 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
) ) )
195, 18mpbird 224 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( ( 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e/ wnel 2572   E!wreu 2672   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   iota_crio 6505   CCcc 8948   RRcr 8949   0cc0 8950   _ici 8952    x. cmul 8955    <_ cle 9081   2c2 10009   RR+crp 10572   ^cexp 11341   Recre 11861   sqrcsqr 11997
This theorem is referenced by:  resqrth  12020  sqrge0  12022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999
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