# 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 ) )`