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Theorem 1idl 26754
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 26744 . . . . 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 4921 . . . . . . . . . 10  |-  ran  G  =  ran  ( 1st `  R
)
72, 6eqtri 2316 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
8 1idl.4 . . . . . . . . 9  |-  U  =  (GId `  H )
95, 7, 8rngolidm 21107 . . . . . . . 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 26749 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( U  e.  I  /\  x  e.  X
) )  ->  ( U H x )  e.  I )
1210, 11eqeltrrd 2371 . . . . . 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 3198 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  U  e.  I )  ->  X  C_  I )
154, 14eqssd 3209 . . 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 21112 . . . 4  |-  ( R  e.  RingOps  ->  U  e.  X
)
1817adantr 451 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  U  e.  X )
19 eleq2 2357 . . 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 1632    e. wcel 1696    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  0rngo  26755  divrngidl  26756  maxidln1  26772
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-pw 3640  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  df-idl 26738
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