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Theorem abvn0b 17717
Description: Another characterization of domains, hinted at in abvtriv 17268: 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 17710 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 abvn0b.b . . . . 5  |-  A  =  (AbsVal `  R )
3 eqid 2462 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2462 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2462 . . . . 5  |-  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  =  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )
6 eqid 2462 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
7 domnrng 17711 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
83, 6, 4domnmuln0 17713 . . . . 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 17267 . . . 4  |-  ( R  e. Domn  ->  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  e.  A )
10 ne0i 3786 . . . 4  |-  ( ( x  e.  ( Base `  R )  |->  if ( x  =  ( 0g
`  R ) ,  0 ,  1 ) )  e.  A  ->  A  =/=  (/) )
119, 10syl 16 . . 3  |-  ( R  e. Domn  ->  A  =/=  (/) )
121, 11jca 532 . 2  |-  ( R  e. Domn  ->  ( R  e. NzRing  /\  A  =/=  (/) ) )
13 n0 3789 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 neanior 2787 . . . . . . . . 9  |-  ( ( y  =/=  ( 0g
`  R )  /\  z  =/=  ( 0g `  R ) )  <->  -.  (
y  =  ( 0g
`  R )  \/  z  =  ( 0g
`  R ) ) )
15 an4 821 . . . . . . . . . . 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 17265 . . . . . . . . . . . 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 1194 . . . . . . . . . . 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 217 . . . . . . . . . 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 437 . . . . . . . . 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 218 . . . . . . . 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 2684 . . . . . . 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 2880 . . . . . 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 1693 . . . . 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 195 . . . 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 569 . . 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 17709 . . 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 212 . 2  |-  ( ( R  e. NzRing  /\  A  =/=  (/) )  ->  R  e. Domn
)
2812, 27impbii 188 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   A.wral 2809   (/)c0 3780   ifcif 3934    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484   Basecbs 14481   .rcmulr 14547   0gc0g 14686  AbsValcabv 17243  NzRingcnzr 17682  Domncdomn 17694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ico 11526  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-mgp 16927  df-rng 16983  df-abv 17244  df-nzr 17683  df-domn 17698
This theorem is referenced by:  nrgdomn  20910
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