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Theorem rngodm1dm2 21085
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 21057 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 eqid 2283 . . . 4  |-  ran  G  =  ran  G
43grpofo 20866 . . 3  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
52, 4syl 15 . 2  |-  ( R  e.  RingOps  ->  G : ( ran  G  X.  ran  G ) -onto-> ran  G )
6 rnplrnml0.1 . . 3  |-  H  =  ( 2nd `  R
)
71, 6, 3rngosm 21048 . 2  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
8 fof 5451 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G :
( ran  G  X.  ran  G ) --> ran  G
)
9 fdm 5393 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
108, 9syl 15 . . 3  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  dom  G  =  ( ran  G  X.  ran  G ) )
11 fdm 5393 . . . 4  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  dom  H  =  ( ran  G  X.  ran  G ) )
12 eqtr 2300 . . . . . . 7  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  G  =  dom  H )
1312dmeqd 4881 . . . . . 6  |-  ( ( dom  G  =  ( ran  G  X.  ran  G )  /\  ( ran 
G  X.  ran  G
)  =  dom  H
)  ->  dom  dom  G  =  dom  dom  H )
1413expcom 424 . . . . 5  |-  ( ( ran  G  X.  ran  G )  =  dom  H  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom 
G  =  dom  dom  H ) )
1514eqcoms 2286 . . . 4  |-  ( dom 
H  =  ( ran 
G  X.  ran  G
)  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom  G  =  dom  dom 
H ) )
1611, 15syl 15 . . 3  |-  ( H : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( dom  G  =  ( ran  G  X.  ran  G )  ->  dom  dom  G  =  dom  dom 
H ) )
1710, 16syl5com 26 . 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 56 1  |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853   RingOpscrngo 21042
This theorem is referenced by:  rngorn1  21086
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-rngo 21043
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