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Metamath Proof Explorer

Theorem absrpcl

Description: The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007) (Proof shortened by Mario Carneiro, 29-May-2016)

Ref Expression
Assertion absrpcl 
|- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )

Proof

Step Hyp Ref Expression
1 absval 
 |-  ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
2 1 adantr 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
3 simpl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> A e. CC )
4 3 cjmulrcld 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( A x. ( * ` A ) ) e. RR )
5 3 cjmulge0d 
 |-  ( ( A e. CC /\ A =/= 0 ) -> 0 <_ ( A x. ( * ` A ) ) )
6 3 cjcld 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( * ` A ) e. CC )
7 simpr 
 |-  ( ( A e. CC /\ A =/= 0 ) -> A =/= 0 )
8 3 7 cjne0d 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( * ` A ) =/= 0 )
9 3 6 7 8 mulne0d 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( A x. ( * ` A ) ) =/= 0 )
10 4 5 9 ne0gt0d 
 |-  ( ( A e. CC /\ A =/= 0 ) -> 0 < ( A x. ( * ` A ) ) )
11 4 10 elrpd 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( A x. ( * ` A ) ) e. RR+ )
12 rpsqrtcl 
 |-  ( ( A x. ( * ` A ) ) e. RR+ -> ( sqrt ` ( A x. ( * ` A ) ) ) e. RR+ )
13 11 12 syl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( sqrt ` ( A x. ( * ` A ) ) ) e. RR+ )
14 2 13 eqeltrd 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )