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Theorem rngolcan 21946
Description: Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringgcl.1  |-  G  =  ( 1st `  R
)
ringgcl.2  |-  X  =  ran  G
Assertion
Ref Expression
rngolcan  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  <->  A  =  B
) )

Proof of Theorem rngolcan
StepHypRef Expression
1 ringgcl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 21939 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringgcl.2 . . 3  |-  X  =  ran  G
43grpolcan 21782 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  <->  A  =  B
) )
52, 4sylan 458 1  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ran crn 4846   ` cfv 5421  (class class class)co 6048   1stc1st 6314   GrpOpcgr 21735   RingOpscrngo 21924
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-1st 6316  df-2nd 6317  df-riota 6516  df-grpo 21740  df-gid 21741  df-ginv 21742  df-ablo 21831  df-rngo 21925
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