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Theorem abv0 15911
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 15901 . . 3  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2435 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
4 abv0.z . . . 4  |-  .0.  =  ( 0g `  R )
53, 4rng0cl 15677 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
62, 5syl 16 . 2  |-  ( F  e.  A  ->  .0.  e.  ( Base `  R
) )
7 eqid 2435 . . 3  |-  .0.  =  .0.
81, 3, 4abveq0 15906 . . 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 1652    e. wcel 1725   ` cfv 5446   0cc0 8982   Basecbs 13461   0gc0g 13715   Ringcrg 15652  AbsValcabv 15896
This theorem is referenced by:  abvdom  15918  abvres  15919  abvcxp  21301  qabvle  21311  ostthlem1  21313  ostth2lem2  21320  ostth3  21324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-grp 14804  df-rng 15655  df-abv 15897
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