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Theorem abv0 15592
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 15582 . . 3  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2284 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
4 abv0.z . . . 4  |-  .0.  =  ( 0g `  R )
53, 4rng0cl 15358 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
62, 5syl 15 . 2  |-  ( F  e.  A  ->  .0.  e.  ( Base `  R
) )
7 eqid 2284 . . 3  |-  .0.  =  .0.
81, 3, 4abveq0 15587 . . 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 1685   ` cfv 5221   0cc0 8733   Basecbs 13144   0gc0g 13396   Ringcrg 15333  AbsValcabv 15577
This theorem is referenced by:  abvdom  15599  abvres  15600  abvcxp  20760  qabvle  20770  ostthlem1  20772  ostth2lem2  20779  ostth3  20783
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-map 6770  df-0g 13400  df-mnd 14363  df-grp 14485  df-rng 15336  df-abv 15578
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