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Theorem eqsqrd 12976
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 12969 . . 3  |-  ( B  e.  CC  ->  E! x  e.  CC  (
( x ^ 2 )  =  B  /\  0  <_  ( Re `  x )  /\  (
_i  x.  x )  e/  RR+ ) )
3 reurmo 3044 . . 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 20 . 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 2651 . . . 4  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
108, 9sylibr 212 . . 3  |-  ( ph  ->  ( _i  x.  A
)  e/  RR+ )
116, 7, 103jca 1168 . 2  |-  ( ph  ->  ( ( A ^
2 )  =  B  /\  0  <_  (
Re `  A )  /\  ( _i  x.  A
)  e/  RR+ ) )
12 sqrcl 12970 . . 3  |-  ( B  e.  CC  ->  ( sqr `  B )  e.  CC )
131, 12syl 16 . 2  |-  ( ph  ->  ( sqr `  B
)  e.  CC )
14 sqrthlem 12971 . . 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 6210 . . . . 5  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
1716eqeq1d 2456 . . . 4  |-  ( x  =  A  ->  (
( x ^ 2 )  =  B  <->  ( A ^ 2 )  =  B ) )
18 fveq2 5802 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
1918breq2d 4415 . . . 4  |-  ( x  =  A  ->  (
0  <_  ( Re `  x )  <->  0  <_  ( Re `  A ) ) )
20 oveq2 6211 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21 neleq1 2790 . . . . 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 1290 . . 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 6210 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( x ^ 2 )  =  ( ( sqr `  B
) ^ 2 ) )
2524eqeq1d 2456 . . . 4  |-  ( x  =  ( sqr `  B
)  ->  ( (
x ^ 2 )  =  B  <->  ( ( sqr `  B ) ^
2 )  =  B ) )
26 fveq2 5802 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( Re `  x )  =  ( Re `  ( sqr `  B ) ) )
2726breq2d 4415 . . . 4  |-  ( x  =  ( sqr `  B
)  ->  ( 0  <_  ( Re `  x )  <->  0  <_  ( Re `  ( sqr `  B ) ) ) )
28 oveq2 6211 . . . . 5  |-  ( x  =  ( sqr `  B
)  ->  ( _i  x.  x )  =  ( _i  x.  ( sqr `  B ) ) )
29 neleq1 2790 . . . . 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 1290 . . 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 3395 . 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 1228 1  |-  ( ph  ->  A  =  ( sqr `  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    e/ wnel 2649   E!wreu 2801   E*wrmo 2802   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   _ici 9398    x. cmul 9401    <_ cle 9533   2c2 10485   RR+crp 11105   ^cexp 11985   Recre 12707   sqrcsqr 12843
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846
This theorem is referenced by:  eqsqr2d  12977  cphsqrcl2  20840
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