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Theorem dvdsr01 15681
Description: In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 16264.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
dvdsr0.b  |-  B  =  ( Base `  R
)
dvdsr0.d  |-  .||  =  (
||r `  R )
dvdsr0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
dvdsr01  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  .|| 
.0.  )

Proof of Theorem dvdsr01
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvdsr0.b . . . . 5  |-  B  =  ( Base `  R
)
2 dvdsr0.z . . . . 5  |-  .0.  =  ( 0g `  R )
31, 2rng0cl 15606 . . . 4  |-  ( R  e.  Ring  ->  .0.  e.  B )
43adantr 452 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .0.  e.  B )
5 eqid 2381 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
61, 5, 2rnglz 15621 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( .r `  R
) X )  =  .0.  )
7 oveq1 6021 . . . . 5  |-  ( x  =  .0.  ->  (
x ( .r `  R ) X )  =  (  .0.  ( .r `  R ) X ) )
87eqeq1d 2389 . . . 4  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) X )  =  .0.  <->  (  .0.  ( .r `  R ) X )  =  .0.  ) )
98rspcev 2989 . . 3  |-  ( (  .0.  e.  B  /\  (  .0.  ( .r `  R ) X )  =  .0.  )  ->  E. x  e.  B  ( x ( .r
`  R ) X )  =  .0.  )
104, 6, 9syl2anc 643 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. x  e.  B  ( x
( .r `  R
) X )  =  .0.  )
11 dvdsr0.d . . . 4  |-  .||  =  (
||r `  R )
121, 11, 5dvdsr2 15673 . . 3  |-  ( X  e.  B  ->  ( X  .||  .0.  <->  E. x  e.  B  ( x
( .r `  R
) X )  =  .0.  ) )
1312adantl 453 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .||  .0.  <->  E. x  e.  B  ( x
( .r `  R
) X )  =  .0.  ) )
1410, 13mpbird 224 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  .|| 
.0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2644   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   Basecbs 13390   .rcmulr 13451   0gc0g 13644   Ringcrg 15581   ||rcdsr 15664
This theorem is referenced by:  ig1pdvds  19960
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 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-riota 6479  df-recs 6563  df-rdg 6598  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-plusg 13463  df-0g 13648  df-mnd 14611  df-grp 14733  df-minusg 14734  df-mgp 15570  df-rng 15584  df-dvdsr 15667
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