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Theorem abvtriv 15602
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 15599 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
Dummy variables  y  z are mutually distinct and distinct from all other variables.
Allowed substitution groups:    A( x)    F( x)

Proof of Theorem abvtriv
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 2286 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
6 drngrng 15515 . 2  |-  ( R  e.  DivRing  ->  R  e.  Ring )
7 biid 229 . . . . 5  |-  ( R  e.  DivRing 
<->  R  e.  DivRing )
8 eldifsn 3752 . . . . 5  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
9 eldifsn 3752 . . . . 5  |-  ( z  e.  ( B  \  {  .0.  } )  <->  ( z  e.  B  /\  z  =/=  .0.  ) )
102, 5, 3drngmcl 15521 . . . . 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 3752 . . . 4  |-  ( ( y ( .r `  R ) z )  e.  ( B  \  {  .0.  } )  <->  ( (
y ( .r `  R ) z )  e.  B  /\  (
y ( .r `  R ) z )  =/=  .0.  ) )
1311, 12sylib 190 . . 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 451 . 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 15601 1  |-  ( R  e.  DivRing  ->  F  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449    \ cdif 3152   ifcif 3568   {csn 3643    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   0cc0 8734   1c1 8735   Basecbs 13144   .rcmulr 13205   0gc0g 13396   DivRingcdr 15508  AbsValcabv 15577
This theorem is referenced by:  ostth1  20778  ostth  20784
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-tpos 6197  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-er 6657  df-map 6771  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-nn 9744  df-2 9801  df-3 9802  df-ico 10658  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-0g 13400  df-mnd 14363  df-grp 14485  df-minusg 14486  df-mgp 15322  df-rng 15336  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-drng 15510  df-abv 15578
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