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Theorem rngomndo 25295
Description: In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
unmnd.1  |-  H  =  ( 2nd `  R
)
Assertion
Ref Expression
rngomndo  |-  ( R  e.  RingOps  ->  H  e. MndOp )

Proof of Theorem rngomndo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 unmnd.1 . . . 4  |-  H  =  ( 2nd `  R
)
3 eqid 2443 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3rngosm 25255 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) )
51, 2, 3rngoass 25261 . . . 4  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R
) ) )  -> 
( ( x H y ) H z )  =  ( x H ( y H z ) ) )
65ralrimivvva 2865 . . 3  |-  ( R  e.  RingOps  ->  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R ) ( ( x H y ) H z )  =  ( x H ( y H z ) ) )
71, 2, 3rngoi 25254 . . . 4  |-  ( R  e.  RingOps  ->  ( ( ( 1st `  R )  e.  AbelOp  /\  H :
( ran  ( 1st `  R )  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R ) )  /\  ( A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R ) ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y ( 1st `  R ) z ) )  =  ( ( x H y ) ( 1st `  R
) ( x H z ) )  /\  ( ( x ( 1st `  R ) y ) H z )  =  ( ( x H z ) ( 1st `  R
) ( y H z ) ) )  /\  E. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
87simprrd 758 . . 3  |-  ( R  e.  RingOps  ->  E. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
92, 1rngorn1 25293 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  dom  dom 
H )
10 xpid11 5214 . . . . . . . 8  |-  ( ( dom  dom  H  X.  dom  dom  H )  =  ( ran  ( 1st `  R )  X.  ran  ( 1st `  R ) )  <->  dom  dom  H  =  ran  ( 1st `  R
) )
1110biimpri 206 . . . . . . 7  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( dom  dom  H  X.  dom  dom  H
)  =  ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) )
12 feq23 5706 . . . . . . 7  |-  ( ( ( dom  dom  H  X.  dom  dom  H )  =  ( ran  ( 1st `  R )  X. 
ran  ( 1st `  R
) )  /\  dom  dom 
H  =  ran  ( 1st `  R ) )  ->  ( H :
( dom  dom  H  X.  dom  dom  H ) --> dom 
dom  H  <->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) ) )
1311, 12mpancom 669 . . . . . 6  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( H :
( dom  dom  H  X.  dom  dom  H ) --> dom 
dom  H  <->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) ) )
14 raleq 3040 . . . . . . . 8  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( A. z  e.  dom  dom  H (
( x H y ) H z )  =  ( x H ( y H z ) )  <->  A. z  e.  ran  ( 1st `  R
) ( ( x H y ) H z )  =  ( x H ( y H z ) ) ) )
1514raleqbi1dv 3048 . . . . . . 7  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( A. y  e.  dom  dom  H A. z  e.  dom  dom  H
( ( x H y ) H z )  =  ( x H ( y H z ) )  <->  A. y  e.  ran  ( 1st `  R
) A. z  e. 
ran  ( 1st `  R
) ( ( x H y ) H z )  =  ( x H ( y H z ) ) ) )
1615raleqbi1dv 3048 . . . . . 6  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( A. x  e.  dom  dom  H A. y  e.  dom  dom  H A. z  e.  dom  dom 
H ( ( x H y ) H z )  =  ( x H ( y H z ) )  <->  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R ) ( ( x H y ) H z )  =  ( x H ( y H z ) ) ) )
17 raleq 3040 . . . . . . 7  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( A. y  e.  dom  dom  H (
( x H y )  =  y  /\  ( y H x )  =  y )  <->  A. y  e.  ran  ( 1st `  R ) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) )
1817rexeqbi1dv 3049 . . . . . 6  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( E. x  e.  dom  dom  H A. y  e.  dom  dom  H
( ( x H y )  =  y  /\  ( y H x )  =  y )  <->  E. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) )
1913, 16, 183anbi123d 1300 . . . . 5  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( ( H : ( dom  dom  H  X.  dom  dom  H
) --> dom  dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom 
H A. z  e. 
dom  dom  H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e.  dom  dom  H A. y  e.  dom  dom  H
( ( x H y )  =  y  /\  ( y H x )  =  y ) )  <->  ( H : ( ran  ( 1st `  R )  X. 
ran  ( 1st `  R
) ) --> ran  ( 1st `  R )  /\  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R
) ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
2019eqcoms 2455 . . . 4  |-  ( ran  ( 1st `  R
)  =  dom  dom  H  ->  ( ( H : ( dom  dom  H  X.  dom  dom  H
) --> dom  dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom 
H A. z  e. 
dom  dom  H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e.  dom  dom  H A. y  e.  dom  dom  H
( ( x H y )  =  y  /\  ( y H x )  =  y ) )  <->  ( H : ( ran  ( 1st `  R )  X. 
ran  ( 1st `  R
) ) --> ran  ( 1st `  R )  /\  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R
) ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
219, 20syl 16 . . 3  |-  ( R  e.  RingOps  ->  ( ( H : ( dom  dom  H  X.  dom  dom  H
) --> dom  dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom 
H A. z  e. 
dom  dom  H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e.  dom  dom  H A. y  e.  dom  dom  H
( ( x H y )  =  y  /\  ( y H x )  =  y ) )  <->  ( H : ( ran  ( 1st `  R )  X. 
ran  ( 1st `  R
) ) --> ran  ( 1st `  R )  /\  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) A. z  e.  ran  ( 1st `  R
) ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
224, 6, 8, 21mpbir3and 1180 . 2  |-  ( R  e.  RingOps  ->  ( H :
( dom  dom  H  X.  dom  dom  H ) --> dom 
dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom  H A. z  e.  dom  dom 
H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
dom  dom  H A. y  e.  dom  dom  H (
( x H y )  =  y  /\  ( y H x )  =  y ) ) )
23 fvex 5866 . . . 4  |-  ( 2nd `  R )  e.  _V
24 eleq1 2515 . . . 4  |-  ( H  =  ( 2nd `  R
)  ->  ( H  e.  _V  <->  ( 2nd `  R
)  e.  _V )
)
2523, 24mpbiri 233 . . 3  |-  ( H  =  ( 2nd `  R
)  ->  H  e.  _V )
26 eqid 2443 . . . 4  |-  dom  dom  H  =  dom  dom  H
2726ismndo1 25218 . . 3  |-  ( H  e.  _V  ->  ( H  e. MndOp  <->  ( H :
( dom  dom  H  X.  dom  dom  H ) --> dom 
dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom  H A. z  e.  dom  dom 
H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
dom  dom  H A. y  e.  dom  dom  H (
( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
282, 25, 27mp2b 10 . 2  |-  ( H  e. MndOp 
<->  ( H : ( dom  dom  H  X.  dom  dom  H ) --> dom 
dom  H  /\  A. x  e.  dom  dom  H A. y  e.  dom  dom  H A. z  e.  dom  dom 
H ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  E. x  e. 
dom  dom  H A. y  e.  dom  dom  H (
( x H y )  =  y  /\  ( y H x )  =  y ) ) )
2922, 28sylibr 212 1  |-  ( R  e.  RingOps  ->  H  e. MndOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    X. cxp 4987   dom cdm 4989   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   AbelOpcablo 25155  MndOpcmndo 25211   RingOpscrngo 25249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786  df-grpo 25065  df-ablo 25156  df-ass 25187  df-exid 25189  df-mgmOLD 25193  df-sgrOLD 25205  df-mndo 25212  df-rngo 25250
This theorem is referenced by:  rngoidmlem  25297  rngo1cl  25303  isdrngo2  30336
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