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Theorem 1idl 30357
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 30347 . . . . 5  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )
43adantr 465 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  I  C_  X )
5 1idl.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
61rneqi 5235 . . . . . . . . . 10  |-  ran  G  =  ran  ( 1st `  R
)
72, 6eqtri 2496 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
8 1idl.4 . . . . . . . . 9  |-  U  =  (GId `  H )
95, 7, 8rngolidm 25257 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( U H x )  =  x )
109ad2ant2rl 748 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  =  x )
111, 5, 2idlrmulcl 30352 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  e.  I )
1210, 11eqeltrrd 2556 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  x  e.  I )
1312expr 615 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  ( x  e.  X  ->  x  e.  I ) )
1413ssrdv 3515 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  X  C_  I )
154, 14eqssd 3526 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  I  =  X )
1615ex 434 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( U  e.  I  ->  I  =  X ) )
177, 5, 8rngo1cl 25262 . . . 4  |-  ( R  e.  RingOps  ->  U  e.  X
)
1817adantr 465 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  U  e.  X )
19 eleq2 2540 . . 3  |-  ( I  =  X  ->  ( U  e.  I  <->  U  e.  X ) )
2018, 19syl5ibrcom 222 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  =  X  ->  U  e.  I )
)
2116, 20impbid 191 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  ( U  e.  I  <->  I  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794  GIdcgi 25020   RingOpscrngo 25208   Idlcidl 30338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6256  df-ov 6298  df-1st 6795  df-2nd 6796  df-grpo 25024  df-gid 25025  df-ablo 25115  df-ass 25146  df-exid 25148  df-mgmOLD 25152  df-sgrOLD 25164  df-mndo 25171  df-rngo 25209  df-idl 30341
This theorem is referenced by:  0rngo  30358  divrngidl  30359  maxidln1  30375
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