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Theorem rngorn1eq 21856
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 2387 . . . 4  |-  ran  G  =  ran  G
41, 2, 3rngosm 21817 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
51, 2, 3rngoi 21816 . . . 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 734 . . . 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 16 . . 3  |-  ( R  e.  RingOps  ->  E. x  e.  ran  G A. y  e.  ran  G ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
8 rngmgmbs4 21853 . . 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 643 . 2  |-  ( R  e.  RingOps  ->  ran  H  =  ran  G )
109eqcomd 2392 1  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    X. cxp 4816   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   AbelOpcablo 21717   RingOpscrngo 21811
This theorem is referenced by:  rngoidmlem  21859  rngo1cl  21865  isdrngo2  26265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-rngo 21812
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