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

Theorem rngoidmlem 21090
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 21088 . . . 4  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 21009 . . . 4  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2283 . . . . . 6  |-  ran  H  =  ran  H
5 uridm.3 . . . . . 6  |-  U  =  (GId `  H )
64, 5cmpidelt 20996 . . . . 5  |-  ( ( H  e.  ( Magma  i^i 
ExId  )  /\  A  e. 
ran  H )  -> 
( ( U H A )  =  A  /\  ( A H U )  =  A ) )
76ex 423 . . . 4  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
82, 3, 73syl 18 . . 3  |-  ( R  e.  RingOps  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
9 eqid 2283 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
101, 9rngorn1eq 21087 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
11 uridm.2 . . . . 5  |-  X  =  ran  ( 1st `  R
)
12 eqtr 2300 . . . . . 6  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  H
)  ->  X  =  ran  H )
13 simpl 443 . . . . . . . . 9  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  ->  X  =  ran  H )
1413eleq2d 2350 . . . . . . . 8  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  -> 
( A  e.  X  <->  A  e.  ran  H ) )
1514imbi1d 308 . . . . . . 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 423 . . . . . 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 15 . . . . 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 651 . . . 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 32 . . 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 223 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
2120imp 418 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985  MndOpcmndo 21004   RingOpscrngo 21042
This theorem is referenced by:  rngolidm  21091  rngoridm  21092
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-reu 2550  df-rmo 2551  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-riota 6304  df-grpo 20858  df-gid 20859  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