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Theorem drngmul0or 16853
Description: A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)
Hypotheses
Ref Expression
drngmuleq0.b  |-  B  =  ( Base `  R
)
drngmuleq0.o  |-  .0.  =  ( 0g `  R )
drngmuleq0.t  |-  .x.  =  ( .r `  R )
drngmuleq0.r  |-  ( ph  ->  R  e.  DivRing )
drngmuleq0.x  |-  ( ph  ->  X  e.  B )
drngmuleq0.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
drngmul0or  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  )
) )

Proof of Theorem drngmul0or
StepHypRef Expression
1 df-ne 2608 . . . . 5  |-  ( X  =/=  .0.  <->  -.  X  =  .0.  )
2 oveq2 6099 . . . . . . . 8  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  ( ( ( invr `  R
) `  X )  .x.  .0.  ) )
32ad2antlr 726 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  ( ( ( invr `  R
) `  X )  .x.  .0.  ) )
4 drngmuleq0.r . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  DivRing )
54adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e.  DivRing )
6 drngmuleq0.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
76adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  B )
8 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  =/= 
.0.  )
9 drngmuleq0.b . . . . . . . . . . . 12  |-  B  =  ( Base `  R
)
10 drngmuleq0.o . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
11 drngmuleq0.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
12 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
13 eqid 2443 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
149, 10, 11, 12, 13drnginvrl 16851 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
155, 7, 8, 14syl3anc 1218 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
1615oveq1d 6106 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( 1r `  R ) 
.x.  Y ) )
17 drngrng 16839 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  R  e.  Ring )
184, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
1918adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e. 
Ring )
209, 10, 13drnginvrcl 16849 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
215, 7, 8, 20syl3anc 1218 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( (
invr `  R ) `  X )  e.  B
)
22 drngmuleq0.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  B )
2322adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  B )
249, 11rngass 16661 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  (
( ( invr `  R
) `  X )  e.  B  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( ( (
invr `  R ) `  X )  .x.  X
)  .x.  Y )  =  ( ( (
invr `  R ) `  X )  .x.  ( X  .x.  Y ) ) )
2519, 21, 7, 23, 24syl13anc 1220 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) ) )
269, 11, 12rnglidm 16668 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2718, 22, 26syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
2827adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( 1r `  R ) 
.x.  Y )  =  Y )
2916, 25, 283eqtr3d 2483 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3029adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3118adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  R  e.  Ring )
3231adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  R  e.  Ring )
3321adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
349, 11, 10rngrz 16682 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  X )  e.  B
)  ->  ( (
( invr `  R ) `  X )  .x.  .0.  )  =  .0.  )
3532, 33, 34syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  .0.  )  =  .0.  )
363, 30, 353eqtr3d 2483 . . . . . 6  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  Y  =  .0.  )
3736ex 434 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =/=  .0.  ->  Y  =  .0.  ) )
381, 37syl5bir 218 . . . 4  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( -.  X  =  .0.  ->  Y  =  .0.  ) )
3938orrd 378 . . 3  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =  .0.  \/  Y  =  .0.  ) )
4039ex 434 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0. 
->  ( X  =  .0. 
\/  Y  =  .0.  ) ) )
419, 11, 10rnglz 16681 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
4218, 22, 41syl2anc 661 . . . 4  |-  ( ph  ->  (  .0.  .x.  Y
)  =  .0.  )
43 oveq1 6098 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
4443eqeq1d 2451 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
4542, 44syl5ibrcom 222 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
469, 11, 10rngrz 16682 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
4718, 6, 46syl2anc 661 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
48 oveq2 6099 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
4948eqeq1d 2451 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
5047, 49syl5ibrcom 222 . . 3  |-  ( ph  ->  ( Y  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
5145, 50jaod 380 . 2  |-  ( ph  ->  ( ( X  =  .0.  \/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
5240, 51impbid 191 1  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   ` cfv 5418  (class class class)co 6091   Basecbs 14174   .rcmulr 14239   0gc0g 14378   1rcur 16603   Ringcrg 16645   invrcinvr 16763   DivRingcdr 16832
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834
This theorem is referenced by:  drngmulne0  16854  drngmuleq0  16855
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