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Metamath Proof Explorer

Theorem sqrtthlem

Description: Lemma for sqrtth . (Contributed by Mario Carneiro, 10-Jul-2013)

Ref Expression
Assertion sqrtthlem 
|- ( A e. CC -> ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) )

Proof

Step Hyp Ref Expression
1 sqrtval 
 |-  ( A e. CC -> ( sqrt ` A ) = ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) )
2 1 eqcomd 
 |-  ( A e. CC -> ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( sqrt ` A ) )
3 sqrtcl 
 |-  ( A e. CC -> ( sqrt ` A ) e. CC )
4 sqreu 
 |-  ( A e. CC -> E! x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )
5 oveq1 
 |-  ( x = ( sqrt ` A ) -> ( x ^ 2 ) = ( ( sqrt ` A ) ^ 2 ) )
6 5 eqeq1d 
 |-  ( x = ( sqrt ` A ) -> ( ( x ^ 2 ) = A <-> ( ( sqrt ` A ) ^ 2 ) = A ) )
7 fveq2 
 |-  ( x = ( sqrt ` A ) -> ( Re ` x ) = ( Re ` ( sqrt ` A ) ) )
8 7 breq2d 
 |-  ( x = ( sqrt ` A ) -> ( 0 <_ ( Re ` x ) <-> 0 <_ ( Re ` ( sqrt ` A ) ) ) )
9 oveq2 
 |-  ( x = ( sqrt ` A ) -> ( _i x. x ) = ( _i x. ( sqrt ` A ) ) )
10 neleq1 
 |-  ( ( _i x. x ) = ( _i x. ( sqrt ` A ) ) -> ( ( _i x. x ) e/ RR+ <-> ( _i x. ( sqrt ` A ) ) e/ RR+ ) )
11 9 10 syl 
 |-  ( x = ( sqrt ` A ) -> ( ( _i x. x ) e/ RR+ <-> ( _i x. ( sqrt ` A ) ) e/ RR+ ) )
12 6 8 11 3anbi123d 
 |-  ( x = ( sqrt ` A ) -> ( ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) <-> ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) ) )
13 12 riota2 
 |-  ( ( ( sqrt ` A ) e. CC /\ E! x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) -> ( ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) <-> ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( sqrt ` A ) ) )
14 3 4 13 syl2anc 
 |-  ( A e. CC -> ( ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) <-> ( iota_ x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) ) = ( sqrt ` A ) ) )
15 2 14 mpbird 
 |-  ( A e. CC -> ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) )