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Theorem rngodm1dm2 21854
Description: In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
rnplrnml0.1  |-  H  =  ( 2nd `  R
)
rnplrnml0.2  |-  G  =  ( 1st `  R
)
Assertion
Ref Expression
rngodm1dm2  |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )

Proof of Theorem rngodm1dm2
StepHypRef Expression
1 rnplrnml0.2 . . . 4  |-  G  =  ( 1st `  R
)
21rngogrpo 21826 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2387 . . . 4  |-  ran  G  =  ran  G
43grpofo 21635 . . 3  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
52, 4syl 16 . 2  |-  ( R  e.  RingOps  ->  G : ( ran  G  X.  ran  G ) -onto-> ran  G )
6 rnplrnml0.1 . . 3  |-  H  =  ( 2nd `  R
)
71, 6, 3rngosm 21817 . 2  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
8 fof 5593 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
9 fdm 5535 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
108, 9syl 16 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
11 fdm 5535 . . . 4  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  H  =  ( ran  G  X.  ran  G ) )
12 eqtr 2404 . . . . . . 7  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  G  =  dom  H )
1312dmeqd 5012 . . . . . 6  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  dom  G  =  dom  dom  H )
1413expcom 425 . . . . 5  |-  ( ( ran  G  X.  ran  G )  =  dom  H  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom 
G  =  dom  dom  H ) )
1514eqcoms 2390 . . . 4  |-  ( dom 
H  =  ( ran 
G  X.  ran  G
)  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom  G  =  dom  dom 
H ) )
1611, 15syl 16 . . 3  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom  G  =  dom  dom 
H ) )
1710, 16syl5com 28 . 2  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  dom  G  =  dom  dom  H )
)
185, 7, 17sylc 58 1  |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4816   dom cdm 4818   ran crn 4819   -->wf 5390   -onto->wfo 5392   ` cfv 5394   1stc1st 6286   2ndc2nd 6287   GrpOpcgr 21622   RingOpscrngo 21811
This theorem is referenced by:  rngorn1  21855
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-grpo 21627  df-ablo 21718  df-rngo 21812
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