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Theorem domneq0 16286
Description: In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
domneq0.b  |-  B  =  ( Base `  R
)
domneq0.t  |-  .x.  =  ( .r `  R )
domneq0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
domneq0  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )

Proof of Theorem domneq0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 956 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  e.  B  /\  Y  e.  B )
)
2 domneq0.b . . . . . 6  |-  B  =  ( Base `  R
)
3 domneq0.t . . . . . 6  |-  .x.  =  ( .r `  R )
4 domneq0.z . . . . . 6  |-  .0.  =  ( 0g `  R )
52, 3, 4isdomn 16283 . . . . 5  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
65simprbi 451 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) ) )
763ad2ant1 978 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) )
8 oveq1 6029 . . . . . 6  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2397 . . . . 5  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
10 eqeq1 2395 . . . . . 6  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1110orbi1d 684 . . . . 5  |-  ( x  =  X  ->  (
( x  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  y  =  .0.  ) ) )
129, 11imbi12d 312 . . . 4  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  y )  =  .0.  ->  ( X  =  .0.  \/  y  =  .0.  ) ) ) )
13 oveq2 6030 . . . . . 6  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2397 . . . . 5  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
15 eqeq1 2395 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
1615orbi2d 683 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
1714, 16imbi12d 312 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  ( X  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  Y )  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  ) ) ) )
1812, 17rspc2va 3004 . . 3  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) )  ->  ( ( X 
.x.  Y )  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
191, 7, 18syl2anc 643 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  )
) )
20 domnrng 16285 . . . . . 6  |-  ( R  e. Domn  ->  R  e.  Ring )
21203ad2ant1 978 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  R  e.  Ring )
22 simp3 959 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
232, 3, 4rnglz 15629 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
2421, 22, 23syl2anc 643 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
25 oveq1 6029 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
2625eqeq1d 2397 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
2724, 26syl5ibrcom 214 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  .0.  ->  ( X  .x.  Y )  =  .0.  ) )
28 simp2 958 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
292, 3, 4rngrz 15630 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
3021, 28, 29syl2anc 643 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
31 oveq2 6030 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
3231eqeq1d 2397 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
3330, 32syl5ibrcom 214 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  =  .0.  ->  ( X  .x.  Y )  =  .0.  ) )
3427, 33jaod 370 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  =  .0. 
\/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
3519, 34impbid 184 1  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   ` cfv 5396  (class class class)co 6022   Basecbs 13398   .rcmulr 13459   0gc0g 13652   Ringcrg 15589  NzRingcnzr 16257  Domncdomn 16269
This theorem is referenced by:  domnmuln0  16287  opprdomn  16290  fidomndrnglem  16295  domnchr  16738  znidomb  16767  fta1glem2  19958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-mgp 15578  df-rng 15592  df-nzr 16258  df-domn 16273
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