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Theorem abvn0b 16350
Description: Another characterization of domains, hinted at in abvtriv 15917: 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 16343 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 abvn0b.b . . . . 5  |-  A  =  (AbsVal `  R )
3 eqid 2435 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2435 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2435 . . . . 5  |-  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  =  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )
6 eqid 2435 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
7 domnrng 16344 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
83, 6, 4domnmuln0 16346 . . . . 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 15916 . . . 4  |-  ( R  e. Domn  ->  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  e.  A )
10 ne0i 3626 . . . 4  |-  ( ( x  e.  ( Base `  R )  |->  if ( x  =  ( 0g
`  R ) ,  0 ,  1 ) )  e.  A  ->  A  =/=  (/) )
119, 10syl 16 . . 3  |-  ( R  e. Domn  ->  A  =/=  (/) )
121, 11jca 519 . 2  |-  ( R  e. Domn  ->  ( R  e. NzRing  /\  A  =/=  (/) ) )
13 n0 3629 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 neanior 2683 . . . . . . . . 9  |-  ( ( y  =/=  ( 0g
`  R )  /\  z  =/=  ( 0g `  R ) )  <->  -.  (
y  =  ( 0g
`  R )  \/  z  =  ( 0g
`  R ) ) )
15 an4 798 . . . . . . . . . . 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 15914 . . . . . . . . . . . 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 1156 . . . . . . . . . . 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 209 . . . . . . . . . 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 427 . . . . . . . . 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 210 . . . . . . . 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 2660 . . . . . . 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 2790 . . . . . 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 1644 . . . . 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 188 . . . 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 553 . . 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 16342 . . 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 204 . 2  |-  ( ( R  e. NzRing  /\  A  =/=  (/) )  ->  R  e. Domn
)
2812, 27impbii 181 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   (/)c0 3620   ifcif 3731    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   0cc0 8979   1c1 8980   Basecbs 13457   .rcmulr 13518   0gc0g 13711  AbsValcabv 15892  NzRingcnzr 16316  Domncdomn 16328
This theorem is referenced by:  nrgdomn  18695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-ico 10911  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-plusg 13530  df-0g 13715  df-mnd 14678  df-grp 14800  df-minusg 14801  df-mgp 15637  df-rng 15651  df-abv 15893  df-nzr 16317  df-domn 16332
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