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Theorem abv0 15596
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 15586 . . 3  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
4 abv0.z . . . 4  |-  .0.  =  ( 0g `  R )
53, 4rng0cl 15362 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
62, 5syl 15 . 2  |-  ( F  e.  A  ->  .0.  e.  ( Base `  R
) )
7 eqid 2283 . . 3  |-  .0.  =  .0.
81, 3, 4abveq0 15591 . . 3  |-  ( ( F  e.  A  /\  .0.  e.  ( Base `  R
) )  ->  (
( F `  .0.  )  =  0  <->  .0.  =  .0.  ) )
97, 8mpbiri 224 . 2  |-  ( ( F  e.  A  /\  .0.  e.  ( Base `  R
) )  ->  ( F `  .0.  )  =  0 )
106, 9mpdan 649 1  |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   0cc0 8737   Basecbs 13148   0gc0g 13400   Ringcrg 15337  AbsValcabv 15581
This theorem is referenced by:  abvdom  15603  abvres  15604  abvcxp  20764  qabvle  20774  ostthlem1  20776  ostth2lem2  20783  ostth3  20787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-rng 15340  df-abv 15582
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