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Metamath Proof Explorer

Theorem sumsqeq0

Description: The sum of two squres of reals is zero if and only if both reals are zero. (Contributed by NM, 29-Apr-2005) (Revised by Stefan O'Rear, 5-Oct-2014) (Proof shortened by Mario Carneiro, 28-May-2016)

Ref Expression
Assertion sumsqeq0 
|- ( ( A e. RR /\ B e. RR ) -> ( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 ) )

Proof

Step Hyp Ref Expression
1 resqcl 
 |-  ( A e. RR -> ( A ^ 2 ) e. RR )
2 sqge0 
 |-  ( A e. RR -> 0 <_ ( A ^ 2 ) )
3 1 2 jca 
 |-  ( A e. RR -> ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) )
4 resqcl 
 |-  ( B e. RR -> ( B ^ 2 ) e. RR )
5 sqge0 
 |-  ( B e. RR -> 0 <_ ( B ^ 2 ) )
6 4 5 jca 
 |-  ( B e. RR -> ( ( B ^ 2 ) e. RR /\ 0 <_ ( B ^ 2 ) ) )
7 add20 
 |-  ( ( ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) /\ ( ( B ^ 2 ) e. RR /\ 0 <_ ( B ^ 2 ) ) ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 <-> ( ( A ^ 2 ) = 0 /\ ( B ^ 2 ) = 0 ) ) )
8 3 6 7 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 <-> ( ( A ^ 2 ) = 0 /\ ( B ^ 2 ) = 0 ) ) )
9 recn 
 |-  ( A e. RR -> A e. CC )
10 sqeq0 
 |-  ( A e. CC -> ( ( A ^ 2 ) = 0 <-> A = 0 ) )
11 9 10 syl 
 |-  ( A e. RR -> ( ( A ^ 2 ) = 0 <-> A = 0 ) )
12 recn 
 |-  ( B e. RR -> B e. CC )
13 sqeq0 
 |-  ( B e. CC -> ( ( B ^ 2 ) = 0 <-> B = 0 ) )
14 12 13 syl 
 |-  ( B e. RR -> ( ( B ^ 2 ) = 0 <-> B = 0 ) )
15 11 14 bi2anan9 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( ( A ^ 2 ) = 0 /\ ( B ^ 2 ) = 0 ) <-> ( A = 0 /\ B = 0 ) ) )
16 8 15 bitr2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 ) )