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Theorem rngorn1eq 21103
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 2296 . . . 4  |-  ran  G  =  ran  G
41, 2, 3rngosm 21064 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  G  X.  ran  G ) --> ran  G )
51, 2, 3rngoi 21063 . . . 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 21100 . . 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 2301 1  |-  ( R  e.  RingOps  ->  ran  G  =  ran  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   AbelOpcablo 20964   RingOpscrngo 21058
This theorem is referenced by:  rngoidmlem  21106  rngo1cl  21112  rngodmdmrn  25521  ununr  25523  rngounval2  25528  glmrngo  25585  isdrngo2  26692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059
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