MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngomndo Unicode version

Theorem rngomndo 21088
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 2283 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 unmnd.1 . . . 4  |-  H  =  ( 2nd `  R
)
3 eqid 2283 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3rngosm 21048 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) )
51, 2, 3rngoass 21054 . . . 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 2636 . . 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 21047 . . . . 5  |-  ( 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 ) ) ) )
87simprd 449 . . . 4  |-  ( 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 ) )  /\  ( 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 ) ) )
98simprd 449 . . 3  |-  ( R  e.  RingOps  ->  E. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
102, 1rngorn1 21086 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  dom  dom 
H )
11 xpid11 4900 . . . . . . . 8  |-  ( ( dom  dom  H  X.  dom  dom  H )  =  ( ran  ( 1st `  R )  X.  ran  ( 1st `  R ) )  <->  dom  dom  H  =  ran  ( 1st `  R
) )
1211biimpri 197 . . . . . . 7  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( dom  dom  H  X.  dom  dom  H
)  =  ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) )
13 feq23 5378 . . . . . . 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
) ) )
1412, 13mpancom 650 . . . . . 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
) ) )
15 raleq 2736 . . . . . . . 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 ) ) ) )
1615raleqbi1dv 2744 . . . . . . 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 ) ) ) )
1716raleqbi1dv 2744 . . . . . 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 ) ) ) )
18 raleq 2736 . . . . . . 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 ) ) )
1918rexeqbi1dv 2745 . . . . . 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 ) ) )
2014, 17, 193anbi123d 1252 . . . . 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 ) ) ) )
2120eqcoms 2286 . . . 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 ) ) ) )
2210, 21syl 15 . . 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 ) ) ) )
234, 6, 9, 22mpbir3and 1135 . 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 ) ) )
24 fvex 5539 . . . 4  |-  ( 2nd `  R )  e.  _V
25 eleq1 2343 . . . 4  |-  ( H  =  ( 2nd `  R
)  ->  ( H  e.  _V  <->  ( 2nd `  R
)  e.  _V )
)
2624, 25mpbiri 224 . . 3  |-  ( H  =  ( 2nd `  R
)  ->  H  e.  _V )
27 eqid 2283 . . . 4  |-  dom  dom  H  =  dom  dom  H
2827ismndo1 21011 . . 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 ) ) ) )
292, 26, 28mp2b 9 . 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 ) ) )
3023, 29sylibr 203 1  |-  ( R  e.  RingOps  ->  H  e. MndOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   AbelOpcablo 20948  MndOpcmndo 21004   RingOpscrngo 21042
This theorem is referenced by:  rngoidmlem  21090  rngo1cl  21096  ununr  25420  glmrngo  25482  isdrngo2  26589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
  Copyright terms: Public domain W3C validator