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Theorem abvtriv 15856
Description: The trivial absolute value. (This theorem is true as long as  R is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 15853 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
abvtriv.a  |-  A  =  (AbsVal `  R )
abvtriv.b  |-  B  =  ( Base `  R
)
abvtriv.z  |-  .0.  =  ( 0g `  R )
abvtriv.f  |-  F  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  1 ) )
Assertion
Ref Expression
abvtriv  |-  ( R  e.  DivRing  ->  F  e.  A
)
Distinct variable groups:    x,  .0.    x, R    x, B
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem abvtriv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvtriv.a . 2  |-  A  =  (AbsVal `  R )
2 abvtriv.b . 2  |-  B  =  ( Base `  R
)
3 abvtriv.z . 2  |-  .0.  =  ( 0g `  R )
4 abvtriv.f . 2  |-  F  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  1 ) )
5 eqid 2387 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
6 drngrng 15769 . 2  |-  ( R  e.  DivRing  ->  R  e.  Ring )
7 biid 228 . . . . 5  |-  ( R  e.  DivRing 
<->  R  e.  DivRing )
8 eldifsn 3870 . . . . 5  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
9 eldifsn 3870 . . . . 5  |-  ( z  e.  ( B  \  {  .0.  } )  <->  ( z  e.  B  /\  z  =/=  .0.  ) )
102, 5, 3drngmcl 15775 . . . . 5  |-  ( ( R  e.  DivRing  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( y ( .r
`  R ) z )  e.  ( B 
\  {  .0.  }
) )
117, 8, 9, 10syl3anbr 1228 . . . 4  |-  ( ( R  e.  DivRing  /\  (
y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/=  .0.  ) )  ->  (
y ( .r `  R ) z )  e.  ( B  \  {  .0.  } ) )
12 eldifsn 3870 . . . 4  |-  ( ( y ( .r `  R ) z )  e.  ( B  \  {  .0.  } )  <->  ( (
y ( .r `  R ) z )  e.  B  /\  (
y ( .r `  R ) z )  =/=  .0.  ) )
1311, 12sylib 189 . . 3  |-  ( ( R  e.  DivRing  /\  (
y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/=  .0.  ) )  ->  (
( y ( .r
`  R ) z )  e.  B  /\  ( y ( .r
`  R ) z )  =/=  .0.  )
)
1413simprd 450 . 2  |-  ( ( R  e.  DivRing  /\  (
y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/=  .0.  ) )  ->  (
y ( .r `  R ) z )  =/=  .0.  )
151, 2, 3, 4, 5, 6, 14abvtrivd 15855 1  |-  ( R  e.  DivRing  ->  F  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260   ifcif 3682   {csn 3757    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   0cc0 8923   1c1 8924   Basecbs 13396   .rcmulr 13457   0gc0g 13650   DivRingcdr 15762  AbsValcabv 15831
This theorem is referenced by:  ostth1  21194  ostth  21200
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-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  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-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-ico 10854  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-dvr 15715  df-drng 15764  df-abv 15832
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