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Theorem domneq0 18158
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 996 . . 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 18155 . . . . 5  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
65simprbi 462 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) ) )
763ad2ant1 1018 . . 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 6241 . . . . . 6  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2404 . . . . 5  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
10 eqeq1 2406 . . . . . 6  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
1110orbi1d 701 . . . . 5  |-  ( x  =  X  ->  (
( x  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  y  =  .0.  ) ) )
129, 11imbi12d 318 . . . 4  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  y )  =  .0.  ->  ( X  =  .0.  \/  y  =  .0.  ) ) ) )
13 oveq2 6242 . . . . . 6  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2404 . . . . 5  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
15 eqeq1 2406 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
1615orbi2d 700 . . . . 5  |-  ( y  =  Y  ->  (
( X  =  .0. 
\/  y  =  .0.  )  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
1714, 16imbi12d 318 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  ( X  =  .0. 
\/  y  =  .0.  ) )  <->  ( ( X  .x.  Y )  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  ) ) ) )
1812, 17rspc2va 3169 . . 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 659 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .x.  Y
)  =  .0.  ->  ( X  =  .0.  \/  Y  =  .0.  )
) )
20 domnring 18157 . . . . . 6  |-  ( R  e. Domn  ->  R  e.  Ring )
21203ad2ant1 1018 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  R  e.  Ring )
22 simp3 999 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
232, 3, 4ringlz 17447 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
2421, 22, 23syl2anc 659 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
25 oveq1 6241 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
2625eqeq1d 2404 . . . 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 998 . . . . 5  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
292, 3, 4ringrz 17448 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
3021, 28, 29syl2anc 659 . . . 4  |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
31 oveq2 6242 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
3231eqeq1d 2404 . . . 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 378 . 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   ` cfv 5525  (class class class)co 6234   Basecbs 14733   .rcmulr 14802   0gc0g 14946   Ringcrg 17410  NzRingcnzr 18117  Domncdomn 18140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-plusg 14814  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-grp 16273  df-minusg 16274  df-mgp 17354  df-ring 17412  df-nzr 18118  df-domn 18144
This theorem is referenced by:  domnmuln0  18159  opprdomn  18162  fidomndrnglem  18167  domnchr  18761  znidomb  18790  fta1glem2  22751  lidldomn1  38218
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