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Theorem rngorn1 21198
Description: In a unital ring the range 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
rngorn1  |-  ( R  e.  RingOps  ->  ran  G  =  dom  dom  H )

Proof of Theorem rngorn1
StepHypRef Expression
1 rnplrnml0.2 . . . 4  |-  G  =  ( 1st `  R
)
21rngogrpo 21169 . . 3  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 grporndm 20989 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
42, 3syl 15 . 2  |-  ( R  e.  RingOps  ->  ran  G  =  dom  dom  G )
5 rnplrnml0.1 . . 3  |-  H  =  ( 2nd `  R
)
65, 1rngodm1dm2 21197 . 2  |-  ( R  e.  RingOps  ->  dom  dom  G  =  dom  dom  H )
74, 6eqtrd 2390 1  |-  ( R  e.  RingOps  ->  ran  G  =  dom  dom  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   dom cdm 4771   ran crn 4772   ` cfv 5337   1stc1st 6207   2ndc2nd 6208   GrpOpcgr 20965   RingOpscrngo 21154
This theorem is referenced by:  rngomndo  21200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-ov 5948  df-1st 6209  df-2nd 6210  df-grpo 20970  df-ablo 21061  df-rngo 21155
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