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Theorem drngmul0or 17931
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 2627 . . . . 5  |-  ( X  =/=  .0.  <->  -.  X  =  .0.  )
2 oveq2 6313 . . . . . . . 8  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  ( ( ( invr `  R
) `  X )  .x.  .0.  ) )
32ad2antlr 731 . . . . . . 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 466 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e.  DivRing )
6 drngmuleq0.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
76adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  B )
8 simpr 462 . . . . . . . . . . 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 2429 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
13 eqid 2429 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
149, 10, 11, 12, 13drnginvrl 17929 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
155, 7, 8, 14syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
1615oveq1d 6320 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( 1r `  R ) 
.x.  Y ) )
17 drngring 17917 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  R  e.  Ring )
184, 17syl 17 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
1918adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e. 
Ring )
209, 10, 13drnginvrcl 17927 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
215, 7, 8, 20syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( (
invr `  R ) `  X )  e.  B
)
22 drngmuleq0.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  B )
2322adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  B )
249, 11ringass 17732 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) ) )
269, 11, 12ringlidm 17739 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2718, 22, 26syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
2827adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( 1r `  R ) 
.x.  Y )  =  Y )
2916, 25, 283eqtr3d 2478 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3029adantlr 719 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3118adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  R  e.  Ring )
3231adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  R  e.  Ring )
3321adantlr 719 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
349, 11, 10ringrz 17753 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  X )  e.  B
)  ->  ( (
( invr `  R ) `  X )  .x.  .0.  )  =  .0.  )
3532, 33, 34syl2anc 665 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  .0.  )  =  .0.  )
363, 30, 353eqtr3d 2478 . . . . . 6  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  Y  =  .0.  )
3736ex 435 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =/=  .0.  ->  Y  =  .0.  ) )
381, 37syl5bir 221 . . . 4  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( -.  X  =  .0.  ->  Y  =  .0.  ) )
3938orrd 379 . . 3  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =  .0.  \/  Y  =  .0.  ) )
4039ex 435 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0. 
->  ( X  =  .0. 
\/  Y  =  .0.  ) ) )
419, 11, 10ringlz 17752 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
4218, 22, 41syl2anc 665 . . . 4  |-  ( ph  ->  (  .0.  .x.  Y
)  =  .0.  )
43 oveq1 6312 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
4443eqeq1d 2431 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
4542, 44syl5ibrcom 225 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
469, 11, 10ringrz 17753 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
4718, 6, 46syl2anc 665 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
48 oveq2 6313 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
4948eqeq1d 2431 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
5047, 49syl5ibrcom 225 . . 3  |-  ( ph  ->  ( Y  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
5145, 50jaod 381 . 2  |-  ( ph  ->  ( ( X  =  .0.  \/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
5240, 51impbid 193 1  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   ` cfv 5601  (class class class)co 6305   Basecbs 15084   .rcmulr 15153   0gc0g 15297   1rcur 17670   Ringcrg 17715   invrcinvr 17834   DivRingcdr 17910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-drng 17912
This theorem is referenced by:  drngmulne0  17932  drngmuleq0  17933
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