MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngorn1eq Unicode version

Theorem rngorn1eq 21087
Description: In a unital ring the range of the addition equals the range 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
rngorn1eq  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )

Proof of Theorem rngorn1eq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnplrnml0.2 . . . 4  |-  G  =  ( 1st `  R
)
2 rnplrnml0.1 . . . 4  |-  H  =  ( 2nd `  R
)
3 eqid 2283 . . . 4  |-  ran  G  =  ran  G
41, 2, 3rngosm 21048 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
51, 2, 3rngoi 21047 . . . 4  |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H :
( ran  G  X.  ran  G ) --> ran  G
)  /\  ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e. 
ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  (
y H x )  =  y ) ) ) )
6 simprr 733 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H : ( ran 
G  X.  ran  G
) --> ran  G )  /\  ( A. x  e. 
ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  (
x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  (
( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) )  ->  E. x  e.  ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
75, 6syl 15 . . 3  |-  ( R  e.  RingOps  ->  E. x  e.  ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
8 rngmgmbs4 21084 . . 3  |-  ( ( H : ( ran 
G  X.  ran  G
) --> ran  G  /\  E. x  e.  ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  ( y H x )  =  y ) )  ->  ran  H  =  ran  G )
94, 7, 8syl2anc 642 . 2  |-  ( R  e.  RingOps  ->  ran  H  =  ran  G )
109eqcomd 2288 1  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   AbelOpcablo 20948   RingOpscrngo 21042
This theorem is referenced by:  rngoidmlem  21090  rngo1cl  21096  rngodmdmrn  25418  ununr  25420  rngounval2  25425  glmrngo  25482  isdrngo2  26589
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-rngo 21043
  Copyright terms: Public domain W3C validator