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Theorem abvfge0 17789
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 17788 . . . 4  |-  ( F  e.  A  ->  R  e.  Ring )
3 abvf.b . . . . 5  |-  B  =  ( Base `  R
)
4 eqid 2402 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2402 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2402 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
71, 3, 4, 5, 6isabv 17786 . . . 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 17 . . 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 241 . 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 457 1  |-  ( F  e.  A  ->  F : B --> ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394   -->wf 5564   ` cfv 5568  (class class class)co 6277   0cc0 9521    + caddc 9524    x. cmul 9526   +oocpnf 9654    <_ cle 9658   [,)cico 11583   Basecbs 14839   +g cplusg 14907   .rcmulr 14908   0gc0g 15052   Ringcrg 17516  AbsValcabv 17783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-abv 17784
This theorem is referenced by:  abvf  17790  abvge0  17792
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