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Theorem rngoidmlem 22003
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 22001 . . . 4  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 21922 . . . 4  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2435 . . . . . 6  |-  ran  H  =  ran  H
5 uridm.3 . . . . . 6  |-  U  =  (GId `  H )
64, 5cmpidelt 21909 . . . . 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 2435 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
101, 9rngorn1eq 22000 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
11 uridm.2 . . . . 5  |-  X  =  ran  ( 1st `  R
)
12 eqtr 2452 . . . . . 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 2502 . . . . . . . 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 1652    e. wcel 1725    i^i cin 3311   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340  GIdcgi 21767    ExId cexid 21894   Magmacmagm 21898  MndOpcmndo 21917   RingOpscrngo 21955
This theorem is referenced by:  rngolidm  22004  rngoridm  22005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956
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