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Theorem drngmul0or 15776
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 2545 . . . . 5  |-  ( X  =/=  .0.  <->  -.  X  =  .0.  )
2 oveq2 6021 . . . . . . . 8  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  ( ( ( invr `  R
) `  X )  .x.  .0.  ) )
32ad2antlr 708 . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e.  DivRing )
6 drngmuleq0.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
76adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  B )
8 simpr 448 . . . . . . . . . . 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 2380 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
13 eqid 2380 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
149, 10, 11, 12, 13drnginvrl 15774 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
155, 7, 8, 14syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  X )  =  ( 1r `  R ) )
1615oveq1d 6028 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( 1r `  R ) 
.x.  Y ) )
17 drngrng 15762 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  R  e.  Ring )
184, 17syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
1918adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  R  e. 
Ring )
209, 10, 13drnginvrcl 15772 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
215, 7, 8, 20syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( (
invr `  R ) `  X )  e.  B
)
22 drngmuleq0.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  B )
2322adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  B )
249, 11rngass 15600 . . . . . . . . . 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 1186 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( ( invr `  R
) `  X )  .x.  X )  .x.  Y
)  =  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) ) )
269, 11, 12rnglidm 15607 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2718, 22, 26syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R )  .x.  Y
)  =  Y )
2827adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( 1r `  R ) 
.x.  Y )  =  Y )
2916, 25, 283eqtr3d 2420 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3029adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  ( X  .x.  Y
) )  =  Y )
3118adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  R  e.  Ring )
3231adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  R  e.  Ring )
3321adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( invr `  R ) `  X )  e.  B
)
349, 11, 10rngrz 15621 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  X )  e.  B
)  ->  ( (
( invr `  R ) `  X )  .x.  .0.  )  =  .0.  )
3532, 33, 34syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  (
( ( invr `  R
) `  X )  .x.  .0.  )  =  .0.  )
363, 30, 353eqtr3d 2420 . . . . . 6  |-  ( ( ( ph  /\  ( X  .x.  Y )  =  .0.  )  /\  X  =/=  .0.  )  ->  Y  =  .0.  )
3736ex 424 . . . . 5  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =/=  .0.  ->  Y  =  .0.  ) )
381, 37syl5bir 210 . . . 4  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( -.  X  =  .0.  ->  Y  =  .0.  ) )
3938orrd 368 . . 3  |-  ( (
ph  /\  ( X  .x.  Y )  =  .0.  )  ->  ( X  =  .0.  \/  Y  =  .0.  ) )
4039ex 424 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0. 
->  ( X  =  .0. 
\/  Y  =  .0.  ) ) )
419, 11, 10rnglz 15620 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  .x.  Y )  =  .0.  )
4218, 22, 41syl2anc 643 . . . 4  |-  ( ph  ->  (  .0.  .x.  Y
)  =  .0.  )
43 oveq1 6020 . . . . 5  |-  ( X  =  .0.  ->  ( X  .x.  Y )  =  (  .0.  .x.  Y
) )
4443eqeq1d 2388 . . . 4  |-  ( X  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  (  .0.  .x. 
Y )  =  .0.  ) )
4542, 44syl5ibrcom 214 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
469, 11, 10rngrz 15621 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
4718, 6, 46syl2anc 643 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
48 oveq2 6021 . . . . 5  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
4948eqeq1d 2388 . . . 4  |-  ( Y  =  .0.  ->  (
( X  .x.  Y
)  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
5047, 49syl5ibrcom 214 . . 3  |-  ( ph  ->  ( Y  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
5145, 50jaod 370 . 2  |-  ( ph  ->  ( ( X  =  .0.  \/  Y  =  .0.  )  ->  ( X  .x.  Y )  =  .0.  ) )
5240, 51impbid 184 1  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   ` cfv 5387  (class class class)co 6013   Basecbs 13389   .rcmulr 13450   0gc0g 13643   Ringcrg 15580   1rcur 15582   invrcinvr 15696   DivRingcdr 15755
This theorem is referenced by:  drngmulne0  15777  drngmuleq0  15778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-drng 15757
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