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Theorem rngoidmlem 21106
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 21104 . . . 4  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 21025 . . . 4  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2296 . . . . . 6  |-  ran  H  =  ran  H
5 uridm.3 . . . . . 6  |-  U  =  (GId `  H )
64, 5cmpidelt 21012 . . . . 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 2296 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
101, 9rngorn1eq 21103 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
11 uridm.2 . . . . 5  |-  X  =  ran  ( 1st `  R
)
12 eqtr 2313 . . . . . 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 2363 . . . . . . . 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 1632    e. wcel 1696    i^i cin 3164   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870    ExId cexid 20997   Magmacmagm 21001  MndOpcmndo 21020   RingOpscrngo 21058
This theorem is referenced by:  rngolidm  21107  rngoridm  21108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059
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