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Theorem rngoidl 26752
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 3210 . . 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 2296 . . 3  |-  (GId `  G )  =  (GId
`  G )
63, 4, 5rngo0cl 21081 . 2  |-  ( R  e.  RingOps  ->  (GId `  G
)  e.  X )
73, 4rngogcl 21074 . . . . . 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 2639 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. y  e.  X  ( x G y )  e.  X )
10 eqid 2296 . . . . . . . . 9  |-  ( 2nd `  R )  =  ( 2nd `  R )
113, 10, 4rngocl 21065 . . . . . . . 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 21065 . . . . . . 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 2639 . . . 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 2639 . 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 26742 . 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 1632    e. wcel 1696   A.wral 2556    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:  divrngidl  26756  igenval  26789  igenidl  26791
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-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-rngo 21059  df-idl 26738
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