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Theorem abvn0b 16043
Description: Another characterization of domains, hinted at in abvtriv 15606: 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 16036 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 abvn0b.b . . . . 5  |-  A  =  (AbsVal `  R )
3 eqid 2283 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2283 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2283 . . . . 5  |-  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  =  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )
6 eqid 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
7 domnrng 16037 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
83, 6, 4domnmuln0 16039 . . . . 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 15605 . . . 4  |-  ( R  e. Domn  ->  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  e.  A )
10 ne0i 3461 . . . 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 3464 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 neanior 2531 . . . . . . . . 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 15603 . . . . . . . . . . . 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 2508 . . . . . . 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 2635 . . . . . 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 1666 . . . . 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 16035 . . 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 1684    =/= wne 2446   A.wral 2543   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   Basecbs 13148   .rcmulr 13209   0gc0g 13400  AbsValcabv 15581  NzRingcnzr 16009  Domncdomn 16021
This theorem is referenced by:  nrgdomn  18182
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ico 10662  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-abv 15582  df-nzr 16010  df-domn 16025
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