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Theorem abv0 15846
Description: The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abv0.a  |-  A  =  (AbsVal `  R )
abv0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
abv0  |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )

Proof of Theorem abv0
StepHypRef Expression
1 abv0.a . . . 4  |-  A  =  (AbsVal `  R )
21abvrcl 15836 . . 3  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2387 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
4 abv0.z . . . 4  |-  .0.  =  ( 0g `  R )
53, 4rng0cl 15612 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
62, 5syl 16 . 2  |-  ( F  e.  A  ->  .0.  e.  ( Base `  R
) )
7 eqid 2387 . . 3  |-  .0.  =  .0.
81, 3, 4abveq0 15841 . . 3  |-  ( ( F  e.  A  /\  .0.  e.  ( Base `  R
) )  ->  (
( F `  .0.  )  =  0  <->  .0.  =  .0.  ) )
97, 8mpbiri 225 . 2  |-  ( ( F  e.  A  /\  .0.  e.  ( Base `  R
) )  ->  ( F `  .0.  )  =  0 )
106, 9mpdan 650 1  |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5394   0cc0 8923   Basecbs 13396   0gc0g 13650   Ringcrg 15587  AbsValcabv 15831
This theorem is referenced by:  abvdom  15853  abvres  15854  abvcxp  21176  qabvle  21186  ostthlem1  21188  ostth2lem2  21195  ostth3  21199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-map 6956  df-0g 13654  df-mnd 14617  df-grp 14739  df-rng 15590  df-abv 15832
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