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Theorem domneq0 17391
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 987 . . 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 17388 . . . . 5  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
65simprbi 464 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) ) )
763ad2ant1 1009 . . 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 6119 . . . . . 6  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2451 . . . . 5  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
10 eqeq1 2449 . . . . . 6  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1110orbi1d 702 . . . . 5  |-  ( x  =  X  ->  (
( x  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  y  =  .0.  ) ) )
129, 11imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  y )  =  .0.  ->  ( X  =  .0.  \/  y  =  .0.  ) ) ) )
13 oveq2 6120 . . . . . 6  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2451 . . . . 5  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
15 eqeq1 2449 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
1615orbi2d 701 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
1714, 16imbi12d 320 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  ( X  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  Y )  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  ) ) ) )
1812, 17rspc2va 3101 . . 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 661 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  )
) )
20 domnrng 17390 . . . . . 6  |-  ( R  e. Domn  ->  R  e.  Ring )
21203ad2ant1 1009 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  R  e.  Ring )
22 simp3 990 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
232, 3, 4rnglz 16703 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
2421, 22, 23syl2anc 661 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
25 oveq1 6119 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
2625eqeq1d 2451 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
2724, 26syl5ibrcom 222 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  .0.  ->  ( X  .x.  Y )  =  .0.  ) )
28 simp2 989 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
292, 3, 4rngrz 16704 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
3021, 28, 29syl2anc 661 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
31 oveq2 6120 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
3231eqeq1d 2451 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
3330, 32syl5ibrcom 222 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  =  .0.  ->  ( X  .x.  Y )  =  .0.  ) )
3427, 33jaod 380 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  =  .0. 
\/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
3519, 34impbid 191 1  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2736   ` cfv 5439  (class class class)co 6112   Basecbs 14195   .rcmulr 14260   0gc0g 14399   Ringcrg 16667  NzRingcnzr 17361  Domncdomn 17373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-mgp 16614  df-rng 16669  df-nzr 17362  df-domn 17377
This theorem is referenced by:  domnmuln0  17392  opprdomn  17395  fidomndrnglem  17400  domnchr  17985  znidomb  18016  fta1glem2  21660
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