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Theorem rngodm1dm2 21998
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 21970 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2435 . . . 4  |-  ran  G  =  ran  G
43grpofo 21779 . . 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 21961 . 2  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
8 fof 5645 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
9 fdm 5587 . . . 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 5587 . . . 4  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  H  =  ( ran  G  X.  ran  G ) )
12 eqtr 2452 . . . . . . 7  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  G  =  dom  H )
1312dmeqd 5064 . . . . . 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 2438 . . . 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 1652    e. wcel 1725    X. cxp 4868   dom cdm 4870   ran crn 4871   -->wf 5442   -onto->wfo 5444   ` cfv 5446   1stc1st 6339   2ndc2nd 6340   GrpOpcgr 21766   RingOpscrngo 21955
This theorem is referenced by:  rngorn1  21999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-grpo 21771  df-ablo 21862  df-rngo 21956
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