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Theorem abvn0b 17480
Description: Another characterization of domains, hinted at in abvtriv 17032: 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 17473 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 abvn0b.b . . . . 5  |-  A  =  (AbsVal `  R )
3 eqid 2451 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2451 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2451 . . . . 5  |-  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  =  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )
6 eqid 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
7 domnrng 17474 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
83, 6, 4domnmuln0 17476 . . . . 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 17031 . . . 4  |-  ( R  e. Domn  ->  ( x  e.  ( Base `  R
)  |->  if ( x  =  ( 0g `  R ) ,  0 ,  1 ) )  e.  A )
10 ne0i 3741 . . . 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 3744 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 neanior 2773 . . . . . . . . 9  |-  ( ( y  =/=  ( 0g
`  R )  /\  z  =/=  ( 0g `  R ) )  <->  -.  (
y  =  ( 0g
`  R )  \/  z  =  ( 0g
`  R ) ) )
15 an4 820 . . . . . . . . . . 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 17029 . . . . . . . . . . . 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 1191 . . . . . . . . . . 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 2670 . . . . . . 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 2904 . . . . . 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 1689 . . . . 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 17472 . . 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   A.wral 2795   (/)c0 3735   ifcif 3889    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190   0cc0 9383   1c1 9384   Basecbs 14276   .rcmulr 14341   0gc0g 14480  AbsValcabv 17007  NzRingcnzr 17445  Domncdomn 17457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-ico 11407  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-mgp 16697  df-rng 16753  df-abv 17008  df-nzr 17446  df-domn 17461
This theorem is referenced by:  nrgdomn  20368
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