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Theorem 1idl 28751
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 28741 . . . . 5  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_  X )
43adantr 462 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  I  C_  X )
5 1idl.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
61rneqi 5062 . . . . . . . . . 10  |-  ran  G  =  ran  ( 1st `  R
)
72, 6eqtri 2461 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
8 1idl.4 . . . . . . . . 9  |-  U  =  (GId `  H )
95, 7, 8rngolidm 23846 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( U H x )  =  x )
109ad2ant2rl 743 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  =  x )
111, 5, 2idlrmulcl 28746 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  e.  I )
1210, 11eqeltrrd 2516 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  x  e.  I )
1312expr 612 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  ( x  e.  X  ->  x  e.  I ) )
1413ssrdv 3359 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  X  C_  I )
154, 14eqssd 3370 . . 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 23851 . . . 4  |-  ( R  e.  RingOps  ->  U  e.  X
)
1817adantr 462 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  U  e.  X )
19 eleq2 2502 . . 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 1364    e. wcel 1761    C_ wss 3325   ran crn 4837   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575  GIdcgi 23609   RingOpscrngo 23797   Idlcidl 28732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fo 5421  df-fv 5423  df-riota 6049  df-ov 6093  df-1st 6576  df-2nd 6577  df-grpo 23613  df-gid 23614  df-ablo 23704  df-ass 23735  df-exid 23737  df-mgm 23741  df-sgr 23753  df-mndo 23760  df-rngo 23798  df-idl 28735
This theorem is referenced by:  0rngo  28752  divrngidl  28753  maxidln1  28769
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