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Theorem domneq0 16347
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 16344 . . . . 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 6080 . . . . . 6  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2443 . . . . 5  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
10 eqeq1 2441 . . . . . 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 6081 . . . . . 6  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2443 . . . . 5  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
15 eqeq1 2441 . . . . . 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 3051 . . 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 16346 . . . . . 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 15690 . . . . 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 6080 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
2625eqeq1d 2443 . . . 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 15691 . . . . 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 6081 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
3231eqeq1d 2443 . . . 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 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13459   .rcmulr 13520   0gc0g 13713   Ringcrg 15650  NzRingcnzr 16318  Domncdomn 16330
This theorem is referenced by:  domnmuln0  16348  opprdomn  16351  fidomndrnglem  16356  domnchr  16803  znidomb  16832  fta1glem2  20079
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 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-0g 13717  df-mnd 14680  df-grp 14802  df-minusg 14803  df-mgp 15639  df-rng 15653  df-nzr 16319  df-domn 16334
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