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Theorem rngoidmlem 21859
Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
uridm.1  |-  H  =  ( 2nd `  R
)
uridm.2  |-  X  =  ran  ( 1st `  R
)
uridm.3  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngoidmlem  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )

Proof of Theorem rngoidmlem
StepHypRef Expression
1 uridm.1 . . . . 5  |-  H  =  ( 2nd `  R
)
21rngomndo 21857 . . . 4  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 21778 . . . 4  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2387 . . . . . 6  |-  ran  H  =  ran  H
5 uridm.3 . . . . . 6  |-  U  =  (GId `  H )
64, 5cmpidelt 21765 . . . . 5  |-  ( ( H  e.  ( Magma  i^i 
ExId  )  /\  A  e. 
ran  H )  -> 
( ( U H A )  =  A  /\  ( A H U )  =  A ) )
76ex 424 . . . 4  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
82, 3, 73syl 19 . . 3  |-  ( R  e.  RingOps  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
9 eqid 2387 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
101, 9rngorn1eq 21856 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
11 uridm.2 . . . . 5  |-  X  =  ran  ( 1st `  R
)
12 eqtr 2404 . . . . . 6  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  H
)  ->  X  =  ran  H )
13 simpl 444 . . . . . . . . 9  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  ->  X  =  ran  H )
1413eleq2d 2454 . . . . . . . 8  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  -> 
( A  e.  X  <->  A  e.  ran  H ) )
1514imbi1d 309 . . . . . . 7  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  -> 
( ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) )  <->  ( A  e.  ran  H  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) ) ) )
1615ex 424 . . . . . 6  |-  ( X  =  ran  H  -> 
( R  e.  RingOps  -> 
( ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) )  <->  ( A  e.  ran  H  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) ) ) ) )
1712, 16syl 16 . . . . 5  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  H
)  ->  ( R  e.  RingOps  ->  ( ( A  e.  X  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )  <-> 
( A  e.  ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) ) ) )
1811, 17mpan 652 . . . 4  |-  ( ran  ( 1st `  R
)  =  ran  H  ->  ( R  e.  RingOps  -> 
( ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) )  <->  ( A  e.  ran  H  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) ) ) ) )
1910, 18mpcom 34 . . 3  |-  ( R  e.  RingOps  ->  ( ( A  e.  X  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )  <-> 
( A  e.  ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) ) )
208, 19mpbird 224 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
2120imp 419 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    i^i cin 3262   ran crn 4819   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287  GIdcgi 21623    ExId cexid 21750   Magmacmagm 21754  MndOpcmndo 21773   RingOpscrngo 21811
This theorem is referenced by:  rngolidm  21860  rngoridm  21861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-riota 6485  df-grpo 21627  df-gid 21628  df-ablo 21718  df-ass 21749  df-exid 21751  df-mgm 21755  df-sgr 21767  df-mndo 21774  df-rngo 21812
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