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Theorem intidl 30344
Description: The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
intidl  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  e.  ( Idl `  R
) )

Proof of Theorem intidl
Dummy variables  i  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 intssuni 4310 . . . 4  |-  ( C  =/=  (/)  ->  |^| C  C_  U. C )
213ad2ant2 1018 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  U. C )
3 ssel2 3504 . . . . . . . 8  |-  ( ( C  C_  ( Idl `  R )  /\  i  e.  C )  ->  i  e.  ( Idl `  R
) )
4 eqid 2467 . . . . . . . . 9  |-  ( 1st `  R )  =  ( 1st `  R )
5 eqid 2467 . . . . . . . . 9  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
64, 5idlss 30331 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_ 
ran  ( 1st `  R
) )
73, 6sylan2 474 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
i  C_  ran  ( 1st `  R ) )
87anassrs 648 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  i  C_ 
ran  ( 1st `  R
) )
98ralrimiva 2881 . . . . 5  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1093adant2 1015 . . . 4  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
11 unissb 4283 . . . 4  |-  ( U. C  C_  ran  ( 1st `  R )  <->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1210, 11sylibr 212 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  U. C  C_  ran  ( 1st `  R
) )
132, 12sstrd 3519 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  ran  ( 1st `  R
) )
14 eqid 2467 . . . . . . . 8  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
154, 14idl0cl 30333 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  i )
163, 15sylan2 474 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
(GId `  ( 1st `  R ) )  e.  i )
1716anassrs 648 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (GId `  ( 1st `  R
) )  e.  i )
1817ralrimiva 2881 . . . 4  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  (GId `  ( 1st `  R ) )  e.  i )
19 fvex 5882 . . . . 5  |-  (GId `  ( 1st `  R ) )  e.  _V
2019elint2 4295 . . . 4  |-  ( (GId
`  ( 1st `  R
) )  e.  |^| C 
<-> 
A. i  e.  C  (GId `  ( 1st `  R
) )  e.  i )
2118, 20sylibr 212 . . 3  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  |^| C )
22213adant2 1015 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  (GId `  ( 1st `  R ) )  e.  |^| C )
23 vex 3121 . . . . . 6  |-  x  e. 
_V
2423elint2 4295 . . . . 5  |-  ( x  e.  |^| C  <->  A. i  e.  C  x  e.  i )
25 vex 3121 . . . . . . . . . 10  |-  y  e. 
_V
2625elint2 4295 . . . . . . . . 9  |-  ( y  e.  |^| C  <->  A. i  e.  C  y  e.  i )
27 r19.26 2994 . . . . . . . . . . 11  |-  ( A. i  e.  C  (
x  e.  i  /\  y  e.  i )  <->  ( A. i  e.  C  x  e.  i  /\  A. i  e.  C  y  e.  i ) )
284idladdcl 30334 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  y  e.  i
) )  ->  (
x ( 1st `  R
) y )  e.  i )
2928ex 434 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
303, 29sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
( ( x  e.  i  /\  y  e.  i )  ->  (
x ( 1st `  R
) y )  e.  i ) )
3130anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
3231ralimdva 2875 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  ( A. i  e.  C  ( x  e.  i  /\  y  e.  i
)  ->  A. i  e.  C  ( x
( 1st `  R
) y )  e.  i ) )
33 ovex 6320 . . . . . . . . . . . . 13  |-  ( x ( 1st `  R
) y )  e. 
_V
3433elint2 4295 . . . . . . . . . . . 12  |-  ( ( x ( 1st `  R
) y )  e. 
|^| C  <->  A. i  e.  C  ( x
( 1st `  R
) y )  e.  i )
3532, 34syl6ibr 227 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  ( A. i  e.  C  ( x  e.  i  /\  y  e.  i
)  ->  ( x
( 1st `  R
) y )  e. 
|^| C ) )
3627, 35syl5bir 218 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  (
( A. i  e.  C  x  e.  i  /\  A. i  e.  C  y  e.  i )  ->  ( x
( 1st `  R
) y )  e. 
|^| C ) )
3736expdimp 437 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  A. i  e.  C  x  e.  i )  ->  ( A. i  e.  C  y  e.  i  ->  ( x ( 1st `  R
) y )  e. 
|^| C ) )
3826, 37syl5bi 217 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  A. i  e.  C  x  e.  i )  ->  (
y  e.  |^| C  ->  ( x ( 1st `  R ) y )  e.  |^| C ) )
3938ralrimiv 2879 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  A. i  e.  C  x  e.  i )  ->  A. y  e.  |^| C ( x ( 1st `  R
) y )  e. 
|^| C )
40 eqid 2467 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  R )  =  ( 2nd `  R )
414, 40, 5idllmulcl 30335 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
z ( 2nd `  R
) x )  e.  i )
4241anass1rs 805 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
z ( 2nd `  R
) x )  e.  i )
4342ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
4443an32s 802 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
453, 44sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  ( C  C_  ( Idl `  R )  /\  i  e.  C ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
4645an4s 824 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  (
z  e.  ran  ( 1st `  R )  /\  i  e.  C )
)  ->  ( x  e.  i  ->  ( z ( 2nd `  R
) x )  e.  i ) )
4746anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  C )  ->  (
x  e.  i  -> 
( z ( 2nd `  R ) x )  e.  i ) )
4847ralimdva 2875 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( A. i  e.  C  x  e.  i  ->  A. i  e.  C  ( z ( 2nd `  R
) x )  e.  i ) )
4948imp 429 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  A. i  e.  C  x  e.  i )  ->  A. i  e.  C  ( z
( 2nd `  R
) x )  e.  i )
50 ovex 6320 . . . . . . . . . . . 12  |-  ( z ( 2nd `  R
) x )  e. 
_V
5150elint2 4295 . . . . . . . . . . 11  |-  ( ( z ( 2nd `  R
) x )  e. 
|^| C  <->  A. i  e.  C  ( z
( 2nd `  R
) x )  e.  i )
5249, 51sylibr 212 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  A. i  e.  C  x  e.  i )  ->  (
z ( 2nd `  R
) x )  e. 
|^| C )
534, 40, 5idlrmulcl 30336 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
x ( 2nd `  R
) z )  e.  i )
5453anass1rs 805 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
x ( 2nd `  R
) z )  e.  i )
5554ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
5655an32s 802 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
573, 56sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  ( C  C_  ( Idl `  R )  /\  i  e.  C ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
5857an4s 824 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  (
z  e.  ran  ( 1st `  R )  /\  i  e.  C )
)  ->  ( x  e.  i  ->  ( x ( 2nd `  R
) z )  e.  i ) )
5958anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  C )  ->  (
x  e.  i  -> 
( x ( 2nd `  R ) z )  e.  i ) )
6059ralimdva 2875 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( A. i  e.  C  x  e.  i  ->  A. i  e.  C  ( x ( 2nd `  R
) z )  e.  i ) )
6160imp 429 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  A. i  e.  C  x  e.  i )  ->  A. i  e.  C  ( x
( 2nd `  R
) z )  e.  i )
62 ovex 6320 . . . . . . . . . . . 12  |-  ( x ( 2nd `  R
) z )  e. 
_V
6362elint2 4295 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  R
) z )  e. 
|^| C  <->  A. i  e.  C  ( x
( 2nd `  R
) z )  e.  i )
6461, 63sylibr 212 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  A. i  e.  C  x  e.  i )  ->  (
x ( 2nd `  R
) z )  e. 
|^| C )
6552, 64jca 532 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  /\  A. i  e.  C  x  e.  i )  ->  (
( z ( 2nd `  R ) x )  e.  |^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) )
6665an32s 802 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  C  C_  ( Idl `  R ) )  /\  A. i  e.  C  x  e.  i )  /\  z  e. 
ran  ( 1st `  R
) )  ->  (
( z ( 2nd `  R ) x )  e.  |^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) )
6766ralrimiva 2881 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  A. i  e.  C  x  e.  i )  ->  A. z  e.  ran  ( 1st `  R
) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) )
6839, 67jca 532 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  A. i  e.  C  x  e.  i )  ->  ( A. y  e.  |^| C
( x ( 1st `  R ) y )  e.  |^| C  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) )
6968ex 434 . . . . 5  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  ( A. i  e.  C  x  e.  i  ->  ( A. y  e.  |^| C ( x ( 1st `  R ) y )  e.  |^| C  /\  A. z  e. 
ran  ( 1st `  R
) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) ) )
7024, 69syl5bi 217 . . . 4  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  (
x  e.  |^| C  ->  ( A. y  e. 
|^| C ( x ( 1st `  R
) y )  e. 
|^| C  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) ) )
7170ralrimiv 2879 . . 3  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. x  e.  |^| C ( A. y  e.  |^| C ( x ( 1st `  R
) y )  e. 
|^| C  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) )
72713adant2 1015 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  A. x  e.  |^| C ( A. y  e.  |^| C ( x ( 1st `  R
) y )  e. 
|^| C  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) )
734, 40, 5, 14isidl 30329 . . 3  |-  ( R  e.  RingOps  ->  ( |^| C  e.  ( Idl `  R
)  <->  ( |^| C  C_ 
ran  ( 1st `  R
)  /\  (GId `  ( 1st `  R ) )  e.  |^| C  /\  A. x  e.  |^| C ( A. y  e.  |^| C ( x ( 1st `  R ) y )  e.  |^| C  /\  A. z  e. 
ran  ( 1st `  R
) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) ) ) )
74733ad2ant1 1017 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  ( |^| C  e.  ( Idl `  R )  <->  ( |^| C  C_  ran  ( 1st `  R )  /\  (GId `  ( 1st `  R
) )  e.  |^| C  /\  A. x  e. 
|^| C ( A. y  e.  |^| C ( x ( 1st `  R
) y )  e. 
|^| C  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R
) x )  e. 
|^| C  /\  (
x ( 2nd `  R
) z )  e. 
|^| C ) ) ) ) )
7513, 22, 72, 74mpbir3and 1179 1  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  e.  ( Idl `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   (/)c0 3790   U.cuni 4251   |^|cint 4288   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794  GIdcgi 25003   RingOpscrngo 25191   Idlcidl 30322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-idl 30325
This theorem is referenced by:  inidl  30345  igenidl  30378
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