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Theorem rngorn1eq 23912
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 2443 . . . 4  |-  ran  G  =  ran  G
41, 2, 3rngosm 23873 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
51, 2, 3rngoi 23872 . . . 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 756 . . . 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 23909 . . 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 661 . 2  |-  ( R  e.  RingOps  ->  ran  H  =  ran  G )
109eqcomd 2448 1  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    X. cxp 4843   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   AbelOpcablo 23773   RingOpscrngo 23867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fo 5429  df-fv 5431  df-ov 6099  df-1st 6582  df-2nd 6583  df-rngo 23868
This theorem is referenced by:  rngoidmlem  23915  rngo1cl  23921  isdrngo2  28769
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