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Theorem abveq0 17253
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 17248 . . . . . 6  |-  ( F  e.  A  ->  R  e.  Ring )
3 abvf.b . . . . . . 7  |-  B  =  ( Base `  R
)
4 eqid 2462 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2462 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 abveq0.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
71, 3, 4, 5, 6isabv 17246 . . . . . 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 16 . . . . 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 463 . . 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 457 . . . 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 2852 . . 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 16 . 2  |-  ( F  e.  A  ->  A. x  e.  B  ( ( F `  x )  =  0  <->  x  =  .0.  ) )
14 fveq2 5859 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1514eqeq1d 2464 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  =  0  <->  ( F `  X )  =  0 ) )
16 eqeq1 2466 . . . 4  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1715, 16bibi12d 321 . . 3  |-  ( x  =  X  ->  (
( ( F `  x )  =  0  <-> 
x  =  .0.  )  <->  ( ( F `  X
)  =  0  <->  X  =  .0.  ) ) )
1817rspccva 3208 . 2  |-  ( ( A. x  e.  B  ( ( F `  x )  =  0  <-> 
x  =  .0.  )  /\  X  e.  B
)  ->  ( ( F `  X )  =  0  <->  X  =  .0.  ) )
1913, 18sylan 471 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 369    = wceq 1374    e. wcel 1762   A.wral 2809   class class class wbr 4442   -->wf 5577   ` cfv 5581  (class class class)co 6277   0cc0 9483    + caddc 9486    x. cmul 9488   +oocpnf 9616    <_ cle 9620   [,)cico 11522   Basecbs 14481   +g cplusg 14546   .rcmulr 14547   0gc0g 14686   Ringcrg 16981  AbsValcabv 17243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-abv 17244
This theorem is referenced by:  abvne0  17254  abv0  17258  abvmet  20826
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