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Theorem dvdsr02 15438
Description: Only zero is divisible by zero. (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
dvdsr02  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .||  X  <->  X  =  .0.  ) )

Proof of Theorem dvdsr02
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 15362 . . . 4  |-  ( R  e.  Ring  ->  .0.  e.  B )
43adantr 451 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .0.  e.  B )
5 dvdsr0.d . . . 4  |-  .||  =  (
||r `  R )
6 eqid 2283 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
71, 5, 6dvdsr2 15429 . . 3  |-  (  .0. 
e.  B  ->  (  .0.  .||  X  <->  E. x  e.  B  ( x
( .r `  R
)  .0.  )  =  X ) )
84, 7syl 15 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .||  X  <->  E. x  e.  B  ( x
( .r `  R
)  .0.  )  =  X ) )
91, 6, 2rngrz 15378 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  B )  ->  (
x ( .r `  R )  .0.  )  =  .0.  )
109eqeq1d 2291 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  B )  ->  (
( x ( .r
`  R )  .0.  )  =  X  <->  .0.  =  X ) )
11 eqcom 2285 . . . . . 6  |-  (  .0.  =  X  <->  X  =  .0.  )
1210, 11syl6bb 252 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  B )  ->  (
( x ( .r
`  R )  .0.  )  =  X  <->  X  =  .0.  ) )
1312rexbidva 2560 . . . 4  |-  ( R  e.  Ring  ->  ( E. x  e.  B  ( x ( .r `  R )  .0.  )  =  X  <->  E. x  e.  B  X  =  .0.  )
)
14 rnggrp 15346 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
151grpbn0 14511 . . . . 5  |-  ( R  e.  Grp  ->  B  =/=  (/) )
16 r19.9rzv 3548 . . . . 5  |-  ( B  =/=  (/)  ->  ( X  =  .0.  <->  E. x  e.  B  X  =  .0.  )
)
1714, 15, 163syl 18 . . . 4  |-  ( R  e.  Ring  ->  ( X  =  .0.  <->  E. x  e.  B  X  =  .0.  ) )
1813, 17bitr4d 247 . . 3  |-  ( R  e.  Ring  ->  ( E. x  e.  B  ( x ( .r `  R )  .0.  )  =  X  <->  X  =  .0.  ) )
1918adantr 451 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( E. x  e.  B  ( x ( .r
`  R )  .0.  )  =  X  <->  X  =  .0.  ) )
208, 19bitrd 244 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .||  X  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   ||rcdsr 15420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-dvdsr 15423
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