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Theorem abvn0b 16039
Description: Another characterization of domains, hinted at in abvtriv 15602: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
Hypothesis
Ref Expression
abvn0b.b  |-  A  =  (AbsVal `  R )
Assertion
Ref Expression
abvn0b  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A  =/=  (/) ) )

Proof of Theorem abvn0b
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 domnnzr 16032 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 abvn0b.b . . . . 5  |-  A  =  (AbsVal `  R )
3 eqid 2284 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2284 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2284 . . . . 5  |-  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  =  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )
6 eqid 2284 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
7 domnrng 16033 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
83, 6, 4domnmuln0 16035 . . . . 5  |-  ( ( R  e. Domn  /\  (
y  e.  ( Base `  R )  /\  y  =/=  ( 0g `  R
) )  /\  (
z  e.  ( Base `  R )  /\  z  =/=  ( 0g `  R
) ) )  -> 
( y ( .r
`  R ) z )  =/=  ( 0g
`  R ) )
92, 3, 4, 5, 6, 7, 8abvtrivd 15601 . . . 4  |-  ( R  e. Domn  ->  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  e.  A )
10 ne0i 3462 . . . 4  |-  ( ( x  e.  ( Base `  R )  |->  if ( x  =  ( 0g
`  R ) ,  0 ,  1 ) )  e.  A  ->  A  =/=  (/) )
119, 10syl 15 . . 3  |-  ( R  e. Domn  ->  A  =/=  (/) )
121, 11jca 518 . 2  |-  ( R  e. Domn  ->  ( R  e. NzRing  /\  A  =/=  (/) ) )
13 n0 3465 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 neanior 2532 . . . . . . . . 9  |-  ( ( y  =/=  ( 0g
`  R )  /\  z  =/=  ( 0g `  R ) )  <->  -.  (
y  =  ( 0g
`  R )  \/  z  =  ( 0g
`  R ) ) )
15 an4 797 . . . . . . . . . . 11  |-  ( ( ( y  e.  (
Base `  R )  /\  z  e.  ( Base `  R ) )  /\  ( y  =/=  ( 0g `  R
)  /\  z  =/=  ( 0g `  R ) ) )  <->  ( (
y  e.  ( Base `  R )  /\  y  =/=  ( 0g `  R
) )  /\  (
z  e.  ( Base `  R )  /\  z  =/=  ( 0g `  R
) ) ) )
162, 3, 4, 6abvdom 15599 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  ( y  e.  (
Base `  R )  /\  y  =/=  ( 0g `  R ) )  /\  ( z  e.  ( Base `  R
)  /\  z  =/=  ( 0g `  R ) ) )  ->  (
y ( .r `  R ) z )  =/=  ( 0g `  R ) )
17163expib 1154 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
( ( y  e.  ( Base `  R
)  /\  y  =/=  ( 0g `  R ) )  /\  ( z  e.  ( Base `  R
)  /\  z  =/=  ( 0g `  R ) ) )  ->  (
y ( .r `  R ) z )  =/=  ( 0g `  R ) ) )
1815, 17syl5bi 208 . . . . . . . . . 10  |-  ( x  e.  A  ->  (
( ( y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  /\  ( y  =/=  ( 0g `  R
)  /\  z  =/=  ( 0g `  R ) ) )  ->  (
y ( .r `  R ) z )  =/=  ( 0g `  R ) ) )
1918expdimp 426 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ( y  e.  (
Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( (
y  =/=  ( 0g
`  R )  /\  z  =/=  ( 0g `  R ) )  -> 
( y ( .r
`  R ) z )  =/=  ( 0g
`  R ) ) )
2014, 19syl5bir 209 . . . . . . . 8  |-  ( ( x  e.  A  /\  ( y  e.  (
Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( -.  ( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) )  ->  ( y
( .r `  R
) z )  =/=  ( 0g `  R
) ) )
2120necon4bd 2509 . . . . . . 7  |-  ( ( x  e.  A  /\  ( y  e.  (
Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( (
y ( .r `  R ) z )  =  ( 0g `  R )  ->  (
y  =  ( 0g
`  R )  \/  z  =  ( 0g
`  R ) ) ) )
2221ralrimivva 2636 . . . . . 6  |-  ( x  e.  A  ->  A. y  e.  ( Base `  R
) A. z  e.  ( Base `  R
) ( ( y ( .r `  R
) z )  =  ( 0g `  R
)  ->  ( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) ) ) )
2322exlimiv 1667 . . . . 5  |-  ( E. x  x  e.  A  ->  A. y  e.  (
Base `  R ) A. z  e.  ( Base `  R ) ( ( y ( .r
`  R ) z )  =  ( 0g
`  R )  -> 
( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) ) ) )
2413, 23sylbi 187 . . . 4  |-  ( A  =/=  (/)  ->  A. y  e.  ( Base `  R
) A. z  e.  ( Base `  R
) ( ( y ( .r `  R
) z )  =  ( 0g `  R
)  ->  ( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) ) ) )
2524anim2i 552 . . 3  |-  ( ( R  e. NzRing  /\  A  =/=  (/) )  ->  ( R  e. NzRing  /\  A. y  e.  ( Base `  R
) A. z  e.  ( Base `  R
) ( ( y ( .r `  R
) z )  =  ( 0g `  R
)  ->  ( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) ) ) ) )
263, 6, 4isdomn 16031 . . 3  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. y  e.  ( Base `  R ) A. z  e.  ( Base `  R
) ( ( y ( .r `  R
) z )  =  ( 0g `  R
)  ->  ( y  =  ( 0g `  R )  \/  z  =  ( 0g `  R ) ) ) ) )
2725, 26sylibr 203 . 2  |-  ( ( R  e. NzRing  /\  A  =/=  (/) )  ->  R  e. Domn
)
2812, 27impbii 180 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   (/)c0 3456   ifcif 3566    e. cmpt 4078   ` cfv 5221  (class class class)co 5820   0cc0 8733   1c1 8734   Basecbs 13144   .rcmulr 13205   0gc0g 13396  AbsValcabv 15577  NzRingcnzr 16005  Domncdomn 16017
This theorem is referenced by:  nrgdomn  18178
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-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  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-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-ico 10658  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-plusg 13217  df-0g 13400  df-mnd 14363  df-grp 14485  df-minusg 14486  df-mgp 15322  df-rng 15336  df-abv 15578  df-nzr 16006  df-domn 16021
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