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Theorem rngoidl 26649
Description: A ring  R is an  R ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
rngidl.1  |-  G  =  ( 1st `  R
)
rngidl.2  |-  X  =  ran  G
Assertion
Ref Expression
rngoidl  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )

Proof of Theorem rngoidl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3197 . . 3  |-  X  C_  X
21a1i 10 . 2  |-  ( R  e.  RingOps  ->  X  C_  X
)
3 rngidl.1 . . 3  |-  G  =  ( 1st `  R
)
4 rngidl.2 . . 3  |-  X  =  ran  G
5 eqid 2283 . . 3  |-  (GId `  G )  =  (GId
`  G )
63, 4, 5rngo0cl 21065 . 2  |-  ( R  e.  RingOps  ->  (GId `  G
)  e.  X )
73, 4rngogcl 21058 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  e.  X )
873expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  y  e.  X
)  ->  ( x G y )  e.  X )
98ralrimiva 2626 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. y  e.  X  ( x G y )  e.  X )
10 eqid 2283 . . . . . . . . 9  |-  ( 2nd `  R )  =  ( 2nd `  R )
113, 10, 4rngocl 21049 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
12113com23 1157 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
133, 10, 4rngocl 21049 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
x ( 2nd `  R
) z )  e.  X )
1412, 13jca 518 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) )
15143expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  z  e.  X
)  ->  ( (
z ( 2nd `  R
) x )  e.  X  /\  ( x ( 2nd `  R
) z )  e.  X ) )
1615ralrimiva 2626 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. z  e.  X  ( (
z ( 2nd `  R
) x )  e.  X  /\  ( x ( 2nd `  R
) z )  e.  X ) )
179, 16jca 518 . . 3  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( A. y  e.  X  ( x G y )  e.  X  /\  A. z  e.  X  ( ( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) ) )
1817ralrimiva 2626 . 2  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( A. y  e.  X  ( x G y )  e.  X  /\  A. z  e.  X  ( ( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) ) )
193, 10, 4, 5isidl 26639 . 2  |-  ( R  e.  RingOps  ->  ( X  e.  ( Idl `  R
)  <->  ( X  C_  X  /\  (GId `  G
)  e.  X  /\  A. x  e.  X  ( A. y  e.  X  ( x G y )  e.  X  /\  A. z  e.  X  ( ( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) ) ) ) )
202, 6, 18, 19mpbir3and 1135 1  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    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:  divrngidl  26653  igenval  26686  igenidl  26688
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-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-rngo 21043  df-idl 26635
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