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Theorem abveq0 15607
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 15602 . . . . . 6  |-  ( F  e.  A  ->  R  e.  Ring )
3 abvf.b . . . . . . 7  |-  B  =  ( Base `  R
)
4 eqid 2296 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2296 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 abveq0.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
71, 3, 4, 5, 6isabv 15600 . . . . . 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 15 . . . . 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 232 . . . 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 449 . . 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 443 . . . 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 2631 . . 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 15 . 2  |-  ( F  e.  A  ->  A. x  e.  B  ( ( F `  x )  =  0  <->  x  =  .0.  ) )
14 fveq2 5541 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1514eqeq1d 2304 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  =  0  <->  ( F `  X )  =  0 ) )
16 eqeq1 2302 . . . 4  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1715, 16bibi12d 312 . . 3  |-  ( x  =  X  ->  (
( ( F `  x )  =  0  <-> 
x  =  .0.  )  <->  ( ( F `  X
)  =  0  <->  X  =  .0.  ) ) )
1817rspccva 2896 . 2  |-  ( ( A. x  e.  B  ( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  X  e.  B
)  ->  ( ( F `  X )  =  0  <->  X  =  .0.  ) )
1913, 18sylan 457 1  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( ( F `  X )  =  0  <-> 
X  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753    + caddc 8756    x. cmul 8758    +oocpnf 8880    <_ cle 8884   [,)cico 10674   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Ringcrg 15353  AbsValcabv 15597
This theorem is referenced by:  abvne0  15608  abv0  15612  abvmet  18114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-abv 15598
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