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Theorem rngomndo 21104
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 2296 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 unmnd.1 . . . 4  |-  H  =  ( 2nd `  R
)
3 eqid 2296 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3rngosm 21064 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) )
51, 2, 3rngoass 21070 . . . 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 2649 . . 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 21063 . . . . 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 21102 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  dom  dom 
H )
11 xpid11 4916 . . . . . . . 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 5394 . . . . . . 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 2749 . . . . . . . 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 2757 . . . . . . 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 2757 . . . . . 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 2749 . . . . . . 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 2758 . . . . . 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 2299 . . . 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 5555 . . . 4  |-  ( 2nd `  R )  e.  _V
25 eleq1 2356 . . . 4  |-  ( H  =  ( 2nd `  R
)  ->  ( H  e.  _V  <->  ( 2nd `  R
)  e.  _V )
)
2624, 25mpbiri 224 . . 3  |-  ( H  =  ( 2nd `  R
)  ->  H  e.  _V )
27 eqid 2296 . . . 4  |-  dom  dom  H  =  dom  dom  H
2827ismndo1 21027 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   AbelOpcablo 20964  MndOpcmndo 21020   RingOpscrngo 21058
This theorem is referenced by:  rngoidmlem  21106  rngo1cl  21112  ununr  25523  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-grpo 20874  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059
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