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Theorem rngoidl 28836
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 3387 . . 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 2443 . . 3  |-  (GId `  G )  =  (GId
`  G )
63, 4, 5rngo0cl 23897 . 2  |-  ( R  e.  RingOps  ->  (GId `  G
)  e.  X )
73, 4rngogcl 23890 . . . . . 6  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  e.  X )
873expa 1187 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  y  e.  X
)  ->  ( x G y )  e.  X )
98ralrimiva 2811 . . . 4  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  A. y  e.  X  ( x G y )  e.  X )
10 eqid 2443 . . . . . . . . 9  |-  ( 2nd `  R )  =  ( 2nd `  R )
113, 10, 4rngocl 23881 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
12113com23 1193 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X  /\  z  e.  X )  ->  (
z ( 2nd `  R
) x )  e.  X )
133, 10, 4rngocl 23881 . . . . . . 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 1187 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  z  e.  X
)  ->  ( (
z ( 2nd `  R
) x )  e.  X  /\  ( x ( 2nd `  R
) z )  e.  X ) )
1615ralrimiva 2811 . . . 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 2811 . 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 28826 . 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 1171 1  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   ran crn 4853   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588  GIdcgi 23686   RingOpscrngo 23874   Idlcidl 28819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-riota 6064  df-ov 6106  df-1st 6589  df-2nd 6590  df-grpo 23690  df-gid 23691  df-ablo 23781  df-rngo 23875  df-idl 28822
This theorem is referenced by:  divrngidl  28840  igenval  28873  igenidl  28875
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