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Metamath Proof Explorer

Theorem abvfge0

Description: An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014)

Ref Expression
Hypotheses abvf.a 
|- A = ( AbsVal ` R )
abvf.b 
|- B = ( Base ` R )
Assertion abvfge0 
|- ( F e. A -> F : B --> ( 0 [,) +oo ) )

Proof

Step Hyp Ref Expression
1 abvf.a 
 |-  A = ( AbsVal ` R )
2 abvf.b 
 |-  B = ( Base ` R )
3 1 abvrcl 
 |-  ( F e. A -> R e. Ring )
4 eqid 
 |-  ( +g ` R ) = ( +g ` R )
5 eqid 
 |-  ( .r ` R ) = ( .r ` R )
6 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
7 1 2 4 5 6 isabv 
 |-  ( R e. Ring -> ( F e. A <-> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = ( 0g ` R ) ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) ) )
8 3 7 syl 
 |-  ( F e. A -> ( F e. A <-> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = ( 0g ` R ) ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) ) )
9 8 ibi 
 |-  ( F e. A -> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = ( 0g ` R ) ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) )
10 9 simpld 
 |-  ( F e. A -> F : B --> ( 0 [,) +oo ) )