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Theorem abvfge0 15912
Description: An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a  |-  A  =  (AbsVal `  R )
abvf.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
abvfge0  |-  ( F  e.  A  ->  F : B --> ( 0 [,) 
+oo ) )

Proof of Theorem abvfge0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . 5  |-  A  =  (AbsVal `  R )
21abvrcl 15911 . . . 4  |-  ( F  e.  A  ->  R  e.  Ring )
3 abvf.b . . . . 5  |-  B  =  ( Base `  R
)
4 eqid 2438 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2438 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
71, 3, 4, 5, 6isabv 15909 . . . 4  |-  ( 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 ) ) ) ) ) ) )
82, 7syl 16 . . 3  |-  ( 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 ) ) ) ) ) ) )
98ibi 234 . 2  |-  ( 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 ) ) ) ) ) )
109simpld 447 1  |-  ( F  e.  A  ->  F : B --> ( 0 [,) 
+oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214   -->wf 5452   ` cfv 5456  (class class class)co 6083   0cc0 8992    + caddc 8995    x. cmul 8997    +oocpnf 9119    <_ cle 9123   [,)cico 10920   Basecbs 13471   +g cplusg 13531   .rcmulr 13532   0gc0g 13725   Ringcrg 15662  AbsValcabv 15906
This theorem is referenced by:  abvf  15913  abvge0  15915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-abv 15907
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