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Metamath Proof Explorer

Theorem eqsqrt2d

Description: A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015)

Ref Expression
Hypotheses eqsqrtd.1 
|- ( ph -> A e. CC )
eqsqrtd.2 
|- ( ph -> B e. CC )
eqsqrtd.3 
|- ( ph -> ( A ^ 2 ) = B )
eqsqrt2d.4 
|- ( ph -> 0 < ( Re ` A ) )
Assertion eqsqrt2d 
|- ( ph -> A = ( sqrt ` B ) )

Proof

Step Hyp Ref Expression
1 eqsqrtd.1 
 |-  ( ph -> A e. CC )
2 eqsqrtd.2 
 |-  ( ph -> B e. CC )
3 eqsqrtd.3 
 |-  ( ph -> ( A ^ 2 ) = B )
4 eqsqrt2d.4 
 |-  ( ph -> 0 < ( Re ` A ) )
5 0re 
 |-  0 e. RR
6 1 recld 
 |-  ( ph -> ( Re ` A ) e. RR )
7 ltle 
 |-  ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( 0 < ( Re ` A ) -> 0 <_ ( Re ` A ) ) )
8 5 6 7 sylancr 
 |-  ( ph -> ( 0 < ( Re ` A ) -> 0 <_ ( Re ` A ) ) )
9 4 8 mpd 
 |-  ( ph -> 0 <_ ( Re ` A ) )
10 reim 
 |-  ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) )
11 1 10 syl 
 |-  ( ph -> ( Re ` A ) = ( Im ` ( _i x. A ) ) )
12 4 gt0ne0d 
 |-  ( ph -> ( Re ` A ) =/= 0 )
13 11 12 eqnetrrd 
 |-  ( ph -> ( Im ` ( _i x. A ) ) =/= 0 )
14 rpre 
 |-  ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
15 14 reim0d 
 |-  ( ( _i x. A ) e. RR+ -> ( Im ` ( _i x. A ) ) = 0 )
16 15 necon3ai 
 |-  ( ( Im ` ( _i x. A ) ) =/= 0 -> -. ( _i x. A ) e. RR+ )
17 13 16 syl 
 |-  ( ph -> -. ( _i x. A ) e. RR+ )
18 1 2 3 9 17 eqsqrtd 
 |-  ( ph -> A = ( sqrt ` B ) )