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Theorem 1idl 26651
Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
1idl.1  |-  G  =  ( 1st `  R
)
1idl.2  |-  H  =  ( 2nd `  R
)
1idl.3  |-  X  =  ran  G
1idl.4  |-  U  =  (GId `  H )
Assertion
Ref Expression
1idl  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( U  e.  I  <->  I  =  X ) )

Proof of Theorem 1idl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 1idl.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 1idl.3 . . . . . 6  |-  X  =  ran  G
31, 2idlss 26641 . . . . 5  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )
43adantr 451 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  I  C_  X )
5 1idl.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
61rneqi 4905 . . . . . . . . . 10  |-  ran  G  =  ran  ( 1st `  R
)
72, 6eqtri 2303 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
8 1idl.4 . . . . . . . . 9  |-  U  =  (GId `  H )
95, 7, 8rngolidm 21091 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( U H x )  =  x )
109ad2ant2rl 729 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  =  x )
111, 5, 2idlrmulcl 26646 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  e.  I )
1210, 11eqeltrrd 2358 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  x  e.  I )
1312expr 598 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  ( x  e.  X  ->  x  e.  I ) )
1413ssrdv 3185 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  X  C_  I )
154, 14eqssd 3196 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  I  =  X )
1615ex 423 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( U  e.  I  ->  I  =  X ) )
177, 5, 8rngo1cl 21096 . . . 4  |-  ( R  e.  RingOps  ->  U  e.  X
)
1817adantr 451 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  U  e.  X )
19 eleq2 2344 . . 3  |-  ( I  =  X  ->  ( U  e.  I  <->  U  e.  X ) )
2018, 19syl5ibrcom 213 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  =  X  ->  U  e.  I )
)
2116, 20impbid 183 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( U  e.  I  <->  I  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632
This theorem is referenced by:  0rngo  26652  divrngidl  26653  maxidln1  26669
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-pw 3627  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  df-idl 26635
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