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Theorem rngorn1eq 22000
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 2435 . . . 4  |-  ran  G  =  ran  G
41, 2, 3rngosm 21961 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
51, 2, 3rngoi 21960 . . . 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 21997 . . 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 2440 1  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   AbelOpcablo 21861   RingOpscrngo 21955
This theorem is referenced by:  rngoidmlem  22003  rngo1cl  22009  isdrngo2  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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