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Theorem rngoidl 26634
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 3367 . . 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 2436 . . 3  |-  (GId `  G )  =  (GId
`  G )
63, 4, 5rngo0cl 21986 . 2  |-  ( R  e.  RingOps  ->  (GId `  G
)  e.  X )
73, 4rngogcl 21979 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  e.  X )
873expa 1153 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  y  e.  X
)  ->  ( x G y )  e.  X )
98ralrimiva 2789 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. y  e.  X  ( x G y )  e.  X )
10 eqid 2436 . . . . . . . . 9  |-  ( 2nd `  R )  =  ( 2nd `  R )
113, 10, 4rngocl 21970 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
12113com23 1159 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
133, 10, 4rngocl 21970 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
x ( 2nd `  R
) z )  e.  X )
1412, 13jca 519 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
( z ( 2nd `  R ) x )  e.  X  /\  (
x ( 2nd `  R
) z )  e.  X ) )
15143expa 1153 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  z  e.  X
)  ->  ( (
z ( 2nd `  R
) x )  e.  X  /\  ( x ( 2nd `  R
) z )  e.  X ) )
1615ralrimiva 2789 . . . 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 519 . . 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 2789 . 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 26624 . 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 1137 1  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963   Idlcidl 26617
This theorem is referenced by:  divrngidl  26638  igenval  26671  igenidl  26673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-rngo 21964  df-idl 26620
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