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Theorem rngodm1dm2 25243
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 25215 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2467 . . . 4  |-  ran  G  =  ran  G
43grpofo 25024 . . 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 25206 . 2  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
8 fof 5801 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
9 fdm 5741 . . . 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 5741 . . . 4  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  H  =  ( ran  G  X.  ran  G ) )
12 eqtr 2493 . . . . . . 7  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  G  =  dom  H )
1312dmeqd 5211 . . . . . 6  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  dom  G  =  dom  dom  H )
1413expcom 435 . . . . 5  |-  ( ( ran  G  X.  ran  G )  =  dom  H  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom 
G  =  dom  dom  H ) )
1514eqcoms 2479 . . . 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 30 . 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 60 1  |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    X. cxp 5003   dom cdm 5005   ran crn 5006   -->wf 5590   -onto->wfo 5592   ` cfv 5594   1stc1st 6793   2ndc2nd 6794   GrpOpcgr 25011   RingOpscrngo 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-grpo 25016  df-ablo 25107  df-rngo 25201
This theorem is referenced by:  rngorn1  25244
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