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Theorem drngmul0or 17229
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 2664 . . . . 5  |-  ( X  =/=  .0.  <->  -.  X  =  .0.  )
2 oveq2 6293 . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
13 eqid 2467 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
149, 10, 11, 12, 13drnginvrl 17227 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
155, 7, 8, 14syl3anc 1228 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
1615oveq1d 6300 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( 1r `  R ) 
.x.  Y ) )
17 drngrng 17215 . . . . . . . . . . . 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 17225 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
215, 7, 8, 20syl3anc 1228 . . . . . . . . . 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 17028 . . . . . . . . . 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 1230 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) ) )
269, 11, 12rnglidm 17035 . . . . . . . . . . 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 2516 . . . . . . . 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 17049 . . . . . . . 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 2516 . . . . . 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 17048 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
4218, 22, 41syl2anc 661 . . . 4  |-  ( ph  ->  (  .0.  .x.  Y
)  =  .0.  )
43 oveq1 6292 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
4443eqeq1d 2469 . . . 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 17049 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
4718, 6, 46syl2anc 661 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
48 oveq2 6293 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
4948eqeq1d 2469 . . . 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 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   0gc0g 14698   1rcur 16967   Ringcrg 17012   invrcinvr 17133   DivRingcdr 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-drng 17210
This theorem is referenced by:  drngmulne0  17230  drngmuleq0  17231
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