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Theorem rngoidl 30626
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 3518 . . 3  |-  X  C_  X
21a1i 11 . 2  |-  ( R  e.  RingOps  ->  X  C_  X
)
3 rngidl.1 . . 3  |-  G  =  ( 1st `  R
)
4 rngidl.2 . . 3  |-  X  =  ran  G
5 eqid 2457 . . 3  |-  (GId `  G )  =  (GId
`  G )
63, 4, 5rngo0cl 25527 . 2  |-  ( R  e.  RingOps  ->  (GId `  G
)  e.  X )
73, 4rngogcl 25520 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  e.  X )
873expa 1196 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  y  e.  X
)  ->  ( x G y )  e.  X )
98ralrimiva 2871 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. y  e.  X  ( x G y )  e.  X )
10 eqid 2457 . . . . . . . . 9  |-  ( 2nd `  R )  =  ( 2nd `  R )
113, 10, 4rngocl 25511 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
12113com23 1202 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
133, 10, 4rngocl 25511 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
x ( 2nd `  R
) z )  e.  X )
1412, 13jca 532 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) )
15143expa 1196 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  z  e.  X
)  ->  ( (
z ( 2nd `  R
) x )  e.  X  /\  ( x ( 2nd `  R
) z )  e.  X ) )
1615ralrimiva 2871 . . . 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 532 . . 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 2871 . 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 30616 . 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 1179 1  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25316   RingOpscrngo 25504   Idlcidl 30609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6258  df-ov 6299  df-1st 6799  df-2nd 6800  df-grpo 25320  df-gid 25321  df-ablo 25411  df-rngo 25505  df-idl 30612
This theorem is referenced by:  divrngidl  30630  igenval  30663  igenidl  30665
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