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Theorem drngmul0or 15894
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 6125 . . . . . . . 8  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  ( ( ( invr `  R
) `  X )  .x.  .0.  ) )
32ad2antlr 709 . . . . . . 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 453 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e.  DivRing )
6 drngmuleq0.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
76adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  B )
8 simpr 449 . . . . . . . . . . 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 15892 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
155, 7, 8, 14syl3anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
1615oveq1d 6132 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( 1r `  R ) 
.x.  Y ) )
17 drngrng 15880 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  R  e.  Ring )
184, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
1918adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e. 
Ring )
209, 10, 13drnginvrcl 15890 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
215, 7, 8, 20syl3anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( (
invr `  R ) `  X )  e.  B
)
22 drngmuleq0.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  B )
2322adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  B )
249, 11rngass 15718 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) ) )
269, 11, 12rnglidm 15725 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2718, 22, 26syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
2827adantr 453 . . . . . . . . 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 697 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3118adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  R  e.  Ring )
3231adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  R  e.  Ring )
3321adantlr 697 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
349, 11, 10rngrz 15739 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  X )  e.  B
)  ->  ( (
( invr `  R ) `  X )  .x.  .0.  )  =  .0.  )
3532, 33, 34syl2anc 644 . . . . . . 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 425 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =/=  .0.  ->  Y  =  .0.  ) )
381, 37syl5bir 211 . . . 4  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( -.  X  =  .0.  ->  Y  =  .0.  ) )
3938orrd 369 . . 3  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =  .0.  \/  Y  =  .0.  ) )
4039ex 425 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0. 
->  ( X  =  .0. 
\/  Y  =  .0.  ) ) )
419, 11, 10rnglz 15738 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
4218, 22, 41syl2anc 644 . . . 4  |-  ( ph  ->  (  .0.  .x.  Y
)  =  .0.  )
43 oveq1 6124 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
4443eqeq1d 2451 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
4542, 44syl5ibrcom 215 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
469, 11, 10rngrz 15739 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
4718, 6, 46syl2anc 644 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
48 oveq2 6125 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
4948eqeq1d 2451 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
5047, 49syl5ibrcom 215 . . 3  |-  ( ph  ->  ( Y  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
5145, 50jaod 371 . 2  |-  ( ph  ->  ( ( X  =  .0.  \/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
5240, 51impbid 185 1  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   ` cfv 5489  (class class class)co 6117   Basecbs 13507   .rcmulr 13568   0gc0g 13761   Ringcrg 15698   1rcur 15700   invrcinvr 15814   DivRingcdr 15873
This theorem is referenced by:  drngmulne0  15895  drngmuleq0  15896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-tpos 6515  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-ress 13514  df-plusg 13580  df-mulr 13581  df-0g 13765  df-mnd 14728  df-grp 14850  df-minusg 14851  df-mgp 15687  df-rng 15701  df-ur 15703  df-oppr 15766  df-dvdsr 15784  df-unit 15785  df-invr 15815  df-drng 15875
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