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Theorem abv0 15695
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 15685 . . 3  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2358 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
4 abv0.z . . . 4  |-  .0.  =  ( 0g `  R )
53, 4rng0cl 15461 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
62, 5syl 15 . 2  |-  ( F  e.  A  ->  .0.  e.  ( Base `  R
) )
7 eqid 2358 . . 3  |-  .0.  =  .0.
81, 3, 4abveq0 15690 . . 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 1642    e. wcel 1710   ` cfv 5337   0cc0 8827   Basecbs 13245   0gc0g 13499   Ringcrg 15436  AbsValcabv 15680
This theorem is referenced by:  abvdom  15702  abvres  15703  abvcxp  20876  qabvle  20886  ostthlem1  20888  ostth2lem2  20895  ostth3  20899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-map 6862  df-0g 13503  df-mnd 14466  df-grp 14588  df-rng 15439  df-abv 15681
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