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Theorem eqsqrd 12171
Description: A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
Hypotheses
Ref Expression
eqsqrd.1  |-  ( ph  ->  A  e.  CC )
eqsqrd.2  |-  ( ph  ->  B  e.  CC )
eqsqrd.3  |-  ( ph  ->  ( A ^ 2 )  =  B )
eqsqrd.4  |-  ( ph  ->  0  <_  ( Re `  A ) )
eqsqrd.5  |-  ( ph  ->  -.  ( _i  x.  A )  e.  RR+ )
Assertion
Ref Expression
eqsqrd  |-  ( ph  ->  A  =  ( sqr `  B ) )

Proof of Theorem eqsqrd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqsqrd.2 . . 3  |-  ( ph  ->  B  e.  CC )
2 sqreu 12164 . . 3  |-  ( B  e.  CC  ->  E! x  e.  CC  (
( x ^ 2 )  =  B  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
3 reurmo 2923 . . 3  |-  ( E! x  e.  CC  (
( x ^ 2 )  =  B  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ )  ->  E* x  e.  CC (
( x ^ 2 )  =  B  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
41, 2, 33syl 19 . 2  |-  ( ph  ->  E* x  e.  CC ( ( x ^
2 )  =  B  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ ) )
5 eqsqrd.1 . 2  |-  ( ph  ->  A  e.  CC )
6 eqsqrd.3 . . 3  |-  ( ph  ->  ( A ^ 2 )  =  B )
7 eqsqrd.4 . . 3  |-  ( ph  ->  0  <_  ( Re `  A ) )
8 eqsqrd.5 . . . 4  |-  ( ph  ->  -.  ( _i  x.  A )  e.  RR+ )
9 df-nel 2602 . . . 4  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
108, 9sylibr 204 . . 3  |-  ( ph  ->  ( _i  x.  A
)  e/  RR+ )
116, 7, 103jca 1134 . 2  |-  ( ph  ->  ( ( A ^
2 )  =  B  /\  0  <_  (
Re `  A )  /\  ( _i  x.  A
)  e/  RR+ ) )
12 sqrcl 12165 . . 3  |-  ( B  e.  CC  ->  ( sqr `  B )  e.  CC )
131, 12syl 16 . 2  |-  ( ph  ->  ( sqr `  B
)  e.  CC )
14 sqrthlem 12166 . . 3  |-  ( B  e.  CC  ->  (
( ( sqr `  B
) ^ 2 )  =  B  /\  0  <_  ( Re `  ( sqr `  B ) )  /\  ( _i  x.  ( sqr `  B ) )  e/  RR+ )
)
151, 14syl 16 . 2  |-  ( ph  ->  ( ( ( sqr `  B ) ^ 2 )  =  B  /\  0  <_  ( Re `  ( sqr `  B ) )  /\  ( _i  x.  ( sqr `  B
) )  e/  RR+ )
)
16 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
1716eqeq1d 2444 . . . 4  |-  ( x  =  A  ->  (
( x ^ 2 )  =  B  <->  ( A ^ 2 )  =  B ) )
18 fveq2 5728 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
1918breq2d 4224 . . . 4  |-  ( x  =  A  ->  (
0  <_  ( Re `  x )  <->  0  <_  ( Re `  A ) ) )
20 oveq2 6089 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21 neleq1 2699 . . . . 5  |-  ( ( _i  x.  x )  =  ( _i  x.  A )  ->  (
( _i  x.  x
)  e/  RR+  <->  ( _i  x.  A )  e/  RR+ )
)
2220, 21syl 16 . . . 4  |-  ( x  =  A  ->  (
( _i  x.  x
)  e/  RR+  <->  ( _i  x.  A )  e/  RR+ )
)
2317, 19, 223anbi123d 1254 . . 3  |-  ( x  =  A  ->  (
( ( x ^
2 )  =  B  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ )  <->  ( ( A ^ 2 )  =  B  /\  0  <_ 
( Re `  A
)  /\  ( _i  x.  A )  e/  RR+ )
) )
24 oveq1 6088 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( x ^ 2 )  =  ( ( sqr `  B
) ^ 2 ) )
2524eqeq1d 2444 . . . 4  |-  ( x  =  ( sqr `  B
)  ->  ( (
x ^ 2 )  =  B  <->  ( ( sqr `  B ) ^
2 )  =  B ) )
26 fveq2 5728 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( Re `  x )  =  ( Re `  ( sqr `  B ) ) )
2726breq2d 4224 . . . 4  |-  ( x  =  ( sqr `  B
)  ->  ( 0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( sqr `  B ) ) ) )
28 oveq2 6089 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( _i  x.  x )  =  ( _i  x.  ( sqr `  B ) ) )
29 neleq1 2699 . . . . 5  |-  ( ( _i  x.  x )  =  ( _i  x.  ( sqr `  B ) )  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  B ) )  e/  RR+ )
)
3028, 29syl 16 . . . 4  |-  ( x  =  ( sqr `  B
)  ->  ( (
_i  x.  x )  e/  RR+  <->  ( _i  x.  ( sqr `  B ) )  e/  RR+ )
)
3125, 27, 303anbi123d 1254 . . 3  |-  ( x  =  ( sqr `  B
)  ->  ( (
( x ^ 2 )  =  B  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ )  <->  ( (
( sqr `  B
) ^ 2 )  =  B  /\  0  <_  ( Re `  ( sqr `  B ) )  /\  ( _i  x.  ( sqr `  B ) )  e/  RR+ )
) )
3223, 31rmoi 3250 . 2  |-  ( ( E* x  e.  CC ( ( x ^
2 )  =  B  /\  0  <_  (
Re `  x )  /\  ( _i  x.  x
)  e/  RR+ )  /\  ( A  e.  CC  /\  ( ( A ^
2 )  =  B  /\  0  <_  (
Re `  A )  /\  ( _i  x.  A
)  e/  RR+ ) )  /\  ( ( sqr `  B )  e.  CC  /\  ( ( ( sqr `  B ) ^ 2 )  =  B  /\  0  <_  ( Re `  ( sqr `  B ) )  /\  ( _i  x.  ( sqr `  B
) )  e/  RR+ )
) )  ->  A  =  ( sqr `  B
) )
334, 5, 11, 13, 15, 32syl122anc 1193 1  |-  ( ph  ->  A  =  ( sqr `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    e/ wnel 2600   E!wreu 2707   E*wrmo 2708   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   _ici 8992    x. cmul 8995    <_ cle 9121   2c2 10049   RR+crp 10612   ^cexp 11382   Recre 11902   sqrcsqr 12038
This theorem is referenced by:  eqsqr2d  12172  cphsqrcl2  19149
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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