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Theorem rngoidmlem 25563
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 25561 . . . 4  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 25482 . . . 4  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2392 . . . . . 6  |-  ran  H  =  ran  H
5 uridm.3 . . . . . 6  |-  U  =  (GId `  H )
64, 5cmpidelt 25469 . . . . 5  |-  ( ( H  e.  ( Magma  i^i 
ExId  )  /\  A  e. 
ran  H )  -> 
( ( U H A )  =  A  /\  ( A H U )  =  A ) )
76ex 432 . . . 4  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
82, 3, 73syl 20 . . 3  |-  ( R  e.  RingOps  ->  ( A  e. 
ran  H  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
9 eqid 2392 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
101, 9rngorn1eq 25560 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
11 uridm.2 . . . . 5  |-  X  =  ran  ( 1st `  R
)
12 eqtr 2418 . . . . . 6  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  H
)  ->  X  =  ran  H )
13 simpl 455 . . . . . . . . 9  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  ->  X  =  ran  H )
1413eleq2d 2462 . . . . . . . 8  |-  ( ( X  =  ran  H  /\  R  e.  RingOps )  -> 
( A  e.  X  <->  A  e.  ran  H ) )
1514imbi1d 315 . . . . . . 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 432 . . . . . 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 668 . . . 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 36 . . 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 232 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( ( U H A )  =  A  /\  ( A H U )  =  A ) ) )
2120imp 427 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836    i^i cin 3401   ran crn 4927   ` cfv 5509  (class class class)co 6214   1stc1st 6715   2ndc2nd 6716  GIdcgi 25327    ExId cexid 25454   Magmacmagm 25458  MndOpcmndo 25477   RingOpscrngo 25515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fo 5515  df-fv 5517  df-riota 6176  df-ov 6217  df-1st 6717  df-2nd 6718  df-grpo 25331  df-gid 25332  df-ablo 25422  df-ass 25453  df-exid 25455  df-mgmOLD 25459  df-sgrOLD 25471  df-mndo 25478  df-rngo 25516
This theorem is referenced by:  rngolidm  25564  rngoridm  25565
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