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Theorem abveq0 17793
Description: The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a  |-  A  =  (AbsVal `  R )
abvf.b  |-  B  =  ( Base `  R
)
abveq0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
abveq0  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( ( F `  X )  =  0  <-> 
X  =  .0.  )
)

Proof of Theorem abveq0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . . 7  |-  A  =  (AbsVal `  R )
21abvrcl 17788 . . . . . 6  |-  ( F  e.  A  ->  R  e.  Ring )
3 abvf.b . . . . . . 7  |-  B  =  ( Base `  R
)
4 eqid 2402 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2402 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 abveq0.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
71, 3, 4, 5, 6isabv 17786 . . . . . 6  |-  ( R  e.  Ring  ->  ( F  e.  A  <->  ( F : B --> ( 0 [,) +oo )  /\  A. x  e.  B  ( (
( F `  x
)  =  0  <->  x  =  .0.  )  /\  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 . . . . 5  |-  ( F  e.  A  ->  ( F  e.  A  <->  ( F : B --> ( 0 [,) +oo )  /\  A. x  e.  B  ( (
( F `  x
)  =  0  <->  x  =  .0.  )  /\  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 . . . 4  |-  ( F  e.  A  ->  ( F : B --> ( 0 [,) +oo )  /\  A. x  e.  B  ( ( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  A. y  e.  B  ( ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) )  /\  ( F `  ( x ( +g  `  R ) y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) ) ) ) )
109simprd 461 . . 3  |-  ( F  e.  A  ->  A. x  e.  B  ( (
( F `  x
)  =  0  <->  x  =  .0.  )  /\  A. y  e.  B  (
( F `  (
x ( .r `  R ) y ) )  =  ( ( F `  x )  x.  ( F `  y ) )  /\  ( F `  ( x ( +g  `  R
) y ) )  <_  ( ( F `
 x )  +  ( F `  y
) ) ) ) )
11 simpl 455 . . . 4  |-  ( ( ( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  A. y  e.  B  ( ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) )  /\  ( F `  ( x ( +g  `  R ) y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) ) )  ->  ( ( F `  x )  =  0  <->  x  =  .0.  ) )
1211ralimi 2796 . . 3  |-  ( A. x  e.  B  (
( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  A. y  e.  B  ( ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) )  /\  ( F `  ( x ( +g  `  R ) y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) ) )  ->  A. x  e.  B  ( ( F `  x )  =  0  <->  x  =  .0.  ) )
1310, 12syl 17 . 2  |-  ( F  e.  A  ->  A. x  e.  B  ( ( F `  x )  =  0  <->  x  =  .0.  ) )
14 fveq2 5848 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1514eqeq1d 2404 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  =  0  <->  ( F `  X )  =  0 ) )
16 eqeq1 2406 . . . 4  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1715, 16bibi12d 319 . . 3  |-  ( x  =  X  ->  (
( ( F `  x )  =  0  <-> 
x  =  .0.  )  <->  ( ( F `  X
)  =  0  <->  X  =  .0.  ) ) )
1817rspccva 3158 . 2  |-  ( ( A. x  e.  B  ( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  X  e.  B
)  ->  ( ( F `  X )  =  0  <->  X  =  .0.  ) )
1913, 18sylan 469 1  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( ( F `  X )  =  0  <-> 
X  =  .0.  )
)
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:  abvne0  17794  abv0  17798  abvmet  21386
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