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Theorem rngomndo 24045
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 2451 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 unmnd.1 . . . 4  |-  H  =  ( 2nd `  R
)
3 eqid 2451 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
41, 2, 3rngosm 24005 . . 3  |-  ( R  e.  RingOps  ->  H : ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) --> ran  ( 1st `  R
) )
51, 2, 3rngoass 24011 . . . 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 2907 . . 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 24004 . . . . 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 463 . . . 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 463 . . 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 24043 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  dom  dom 
H )
11 xpid11 5161 . . . . . . . 8  |-  ( ( dom  dom  H  X.  dom  dom  H )  =  ( ran  ( 1st `  R )  X.  ran  ( 1st `  R ) )  <->  dom  dom  H  =  ran  ( 1st `  R
) )
1211biimpri 206 . . . . . . 7  |-  ( dom 
dom  H  =  ran  ( 1st `  R )  ->  ( dom  dom  H  X.  dom  dom  H
)  =  ( ran  ( 1st `  R
)  X.  ran  ( 1st `  R ) ) )
13 feq23 5645 . . . . . . 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 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
) ) )
15 raleq 3015 . . . . . . . 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 3023 . . . . . . 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 3023 . . . . . 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 3015 . . . . . . 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 3024 . . . . . 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 1290 . . . . 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 2463 . . . 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 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 ) ) ) )
234, 6, 9, 22mpbir3and 1171 . 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 5801 . . . 4  |-  ( 2nd `  R )  e.  _V
25 eleq1 2523 . . . 4  |-  ( H  =  ( 2nd `  R
)  ->  ( H  e.  _V  <->  ( 2nd `  R
)  e.  _V )
)
2624, 25mpbiri 233 . . 3  |-  ( H  =  ( 2nd `  R
)  ->  H  e.  _V )
27 eqid 2451 . . . 4  |-  dom  dom  H  =  dom  dom  H
2827ismndo1 23968 . . 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 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 ) ) )
3023, 29sylibr 212 1  |-  ( R  e.  RingOps  ->  H  e. MndOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070    X. cxp 4938   dom cdm 4940   ran crn 4941   -->wf 5514   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   AbelOpcablo 23905  MndOpcmndo 23961   RingOpscrngo 23999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-ov 6195  df-1st 6679  df-2nd 6680  df-grpo 23815  df-ablo 23906  df-ass 23937  df-exid 23939  df-mgm 23943  df-sgr 23955  df-mndo 23962  df-rngo 24000
This theorem is referenced by:  rngoidmlem  24047  rngo1cl  24053  isdrngo2  28904
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