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Theorem dvdsr01 13204
Description: In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (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 . . . 4  |-  B  =  ( Base `  R
)
2 dvdsr0.z . . . 4  |-  .0.  =  ( 0g `  R )
31, 2ring0cl 13135 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  B )
4 eqid 2177 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
51, 4, 2ringlz 13153 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( .r `  R
) X )  =  .0.  )
6 oveq1 5879 . . . . 5  |-  ( x  =  .0.  ->  (
x ( .r `  R ) X )  =  (  .0.  ( .r `  R ) X ) )
76eqeq1d 2186 . . . 4  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) X )  =  .0.  <->  (  .0.  ( .r `  R ) X )  =  .0.  ) )
87rspcev 2841 . . 3  |-  ( (  .0.  e.  B  /\  (  .0.  ( .r `  R ) X )  =  .0.  )  ->  E. x  e.  B  ( x ( .r
`  R ) X )  =  .0.  )
93, 5, 8syl2an2r 595 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. x  e.  B  ( x
( .r `  R
) X )  =  .0.  )
101a1i 9 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  B  =  ( Base `  R
) )
11 dvdsr0.d . . . 4  |-  .||  =  (
||r `  R )
1211a1i 9 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .||  =  (
||r `  R ) )
13 ringsrg 13155 . . . 4  |-  ( R  e.  Ring  ->  R  e. SRing
)
1413adantr 276 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e. SRing )
15 eqidd 2178 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( .r `  R )  =  ( .r `  R
) )
16 simpr 110 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
1710, 12, 14, 15, 16dvdsr2d 13195 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .||  .0.  <->  E. x  e.  B  ( x
( .r `  R
) X )  =  .0.  ) )
189, 17mpbird 167 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  .|| 
.0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   Basecbs 12454   .rcmulr 12529   0gc0g 12693  SRingcsrg 13077   Ringcrg 13110   ||rcdsr 13186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-dvdsr 13189
This theorem is referenced by: (None)
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