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Theorem intidl 31966
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 4281 . . . 4  |-  ( C  =/=  (/)  ->  |^| C  C_  U. C )
213ad2ant2 1027 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  U. C )
3 ssel2 3465 . . . . . . . 8  |-  ( ( C  C_  ( Idl `  R )  /\  i  e.  C )  ->  i  e.  ( Idl `  R
) )
4 eqid 2429 . . . . . . . . 9  |-  ( 1st `  R )  =  ( 1st `  R )
5 eqid 2429 . . . . . . . . 9  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
64, 5idlss 31953 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_ 
ran  ( 1st `  R
) )
73, 6sylan2 476 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
i  C_  ran  ( 1st `  R ) )
87anassrs 652 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  i  C_ 
ran  ( 1st `  R
) )
98ralrimiva 2846 . . . . 5  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1093adant2 1024 . . . 4  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
11 unissb 4253 . . . 4  |-  ( U. C  C_  ran  ( 1st `  R )  <->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1210, 11sylibr 215 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  U. C  C_  ran  ( 1st `  R
) )
132, 12sstrd 3480 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  ran  ( 1st `  R
) )
14 eqid 2429 . . . . . . . 8  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
154, 14idl0cl 31955 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  i )
163, 15sylan2 476 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
(GId `  ( 1st `  R ) )  e.  i )
1716anassrs 652 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (GId `  ( 1st `  R
) )  e.  i )
1817ralrimiva 2846 . . . 4  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  (GId `  ( 1st `  R ) )  e.  i )
19 fvex 5891 . . . . 5  |-  (GId `  ( 1st `  R ) )  e.  _V
2019elint2 4265 . . . 4  |-  ( (GId
`  ( 1st `  R
) )  e.  |^| C 
<-> 
A. i  e.  C  (GId `  ( 1st `  R
) )  e.  i )
2118, 20sylibr 215 . . 3  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  |^| C )
22213adant2 1024 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  (GId `  ( 1st `  R ) )  e.  |^| C )
23 vex 3090 . . . . . 6  |-  x  e. 
_V
2423elint2 4265 . . . . 5  |-  ( x  e.  |^| C  <->  A. i  e.  C  x  e.  i )
25 vex 3090 . . . . . . . . . 10  |-  y  e. 
_V
2625elint2 4265 . . . . . . . . 9  |-  ( y  e.  |^| C  <->  A. i  e.  C  y  e.  i )
27 r19.26 2962 . . . . . . . . . . 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 31956 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  y  e.  i
) )  ->  (
x ( 1st `  R
) y )  e.  i )
2928ex 435 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
303, 29sylan2 476 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
( ( x  e.  i  /\  y  e.  i )  ->  (
x ( 1st `  R
) y )  e.  i ) )
3130anassrs 652 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
3231ralimdva 2840 . . . . . . . . . . . 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 6333 . . . . . . . . . . . . 13  |-  ( x ( 1st `  R
) y )  e. 
_V
3433elint2 4265 . . . . . . . . . . . 12  |-  ( ( x ( 1st `  R
) y )  e. 
|^| C  <->  A. i  e.  C  ( x
( 1st `  R
) y )  e.  i )
3532, 34syl6ibr 230 . . . . . . . . . . 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 221 . . . . . . . . . 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 438 . . . . . . . . 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 220 . . . . . . . 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 2844 . . . . . . 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 2429 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  R )  =  ( 2nd `  R )
414, 40, 5idllmulcl 31957 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
z ( 2nd `  R
) x )  e.  i )
4241anass1rs 814 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
z ( 2nd `  R
) x )  e.  i )
4342ex 435 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
4443an32s 811 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
453, 44sylan2 476 . . . . . . . . . . . . . . 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 833 . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 6333 . . . . . . . . . . . 12  |-  ( z ( 2nd `  R
) x )  e. 
_V
5150elint2 4265 . . . . . . . . . . 11  |-  ( ( z ( 2nd `  R
) x )  e. 
|^| C  <->  A. i  e.  C  ( z
( 2nd `  R
) x )  e.  i )
5249, 51sylibr 215 . . . . . . . . . 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 31958 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
x ( 2nd `  R
) z )  e.  i )
5453anass1rs 814 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
x ( 2nd `  R
) z )  e.  i )
5554ex 435 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
5655an32s 811 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
573, 56sylan2 476 . . . . . . . . . . . . . . 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 833 . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 6333 . . . . . . . . . . . 12  |-  ( x ( 2nd `  R
) z )  e. 
_V
6362elint2 4265 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  R
) z )  e. 
|^| C  <->  A. i  e.  C  ( x
( 2nd `  R
) z )  e.  i )
6461, 63sylibr 215 . . . . . . . . . 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 534 . . . . . . . . 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 811 . . . . . . . 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 2846 . . . . . . 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 534 . . . . . 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 435 . . . . 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 220 . . . 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 2844 . . 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 1024 . 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 31951 . . 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 1026 . 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 1188 1  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  e.  ( Idl `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870    =/= wne 2625   A.wral 2782    C_ wss 3442   (/)c0 3767   U.cuni 4222   |^|cint 4258   ran crn 4855   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806  GIdcgi 25760   RingOpscrngo 25948   Idlcidl 31944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-idl 31947
This theorem is referenced by:  inidl  31967  igenidl  32000
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