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Theorem intidl 26757
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 3900 . . . 4  |-  ( C  =/=  (/)  ->  |^| C  C_  U. C )
213ad2ant2 977 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  U. C )
3 ssel2 3188 . . . . . . . 8  |-  ( ( C  C_  ( Idl `  R )  /\  i  e.  C )  ->  i  e.  ( Idl `  R
) )
4 eqid 2296 . . . . . . . . 9  |-  ( 1st `  R )  =  ( 1st `  R )
5 eqid 2296 . . . . . . . . 9  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
64, 5idlss 26744 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_ 
ran  ( 1st `  R
) )
73, 6sylan2 460 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
i  C_  ran  ( 1st `  R ) )
87anassrs 629 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  i  C_ 
ran  ( 1st `  R
) )
98ralrimiva 2639 . . . . 5  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1093adant2 974 . . . 4  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
11 unissb 3873 . . . 4  |-  ( U. C  C_  ran  ( 1st `  R )  <->  A. i  e.  C  i  C_  ran  ( 1st `  R
) )
1210, 11sylibr 203 . . 3  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  U. C  C_  ran  ( 1st `  R
) )
132, 12sstrd 3202 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  C_  ran  ( 1st `  R
) )
14 eqid 2296 . . . . . . . 8  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
154, 14idl0cl 26746 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  i )
163, 15sylan2 460 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
(GId `  ( 1st `  R ) )  e.  i )
1716anassrs 629 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (GId `  ( 1st `  R
) )  e.  i )
1817ralrimiva 2639 . . . 4  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  A. i  e.  C  (GId `  ( 1st `  R ) )  e.  i )
19 fvex 5555 . . . . 5  |-  (GId `  ( 1st `  R ) )  e.  _V
2019elint2 3885 . . . 4  |-  ( (GId
`  ( 1st `  R
) )  e.  |^| C 
<-> 
A. i  e.  C  (GId `  ( 1st `  R
) )  e.  i )
2118, 20sylibr 203 . . 3  |-  ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  ->  (GId `  ( 1st `  R
) )  e.  |^| C )
22213adant2 974 . 2  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  (GId `  ( 1st `  R ) )  e.  |^| C )
23 vex 2804 . . . . . 6  |-  x  e. 
_V
2423elint2 3885 . . . . 5  |-  ( x  e.  |^| C  <->  A. i  e.  C  x  e.  i )
25 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
2625elint2 3885 . . . . . . . . 9  |-  ( y  e.  |^| C  <->  A. i  e.  C  y  e.  i )
27 r19.26 2688 . . . . . . . . . . 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 26747 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  y  e.  i
) )  ->  (
x ( 1st `  R
) y )  e.  i )
2928ex 423 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
303, 29sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  ( C  C_  ( Idl `  R
)  /\  i  e.  C ) )  -> 
( ( x  e.  i  /\  y  e.  i )  ->  (
x ( 1st `  R
) y )  e.  i ) )
3130anassrs 629 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  C  C_  ( Idl `  R
) )  /\  i  e.  C )  ->  (
( x  e.  i  /\  y  e.  i )  ->  ( x
( 1st `  R
) y )  e.  i ) )
3231ralimdva 2634 . . . . . . . . . . . 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 5899 . . . . . . . . . . . . 13  |-  ( x ( 1st `  R
) y )  e. 
_V
3433elint2 3885 . . . . . . . . . . . 12  |-  ( ( x ( 1st `  R
) y )  e. 
|^| C  <->  A. i  e.  C  ( x
( 1st `  R
) y )  e.  i )
3532, 34syl6ibr 218 . . . . . . . . . . 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 209 . . . . . . . . . 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 426 . . . . . . . . 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 208 . . . . . . . 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 2638 . . . . . . 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 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  R )  =  ( 2nd `  R )
414, 40, 5idllmulcl 26748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
z ( 2nd `  R
) x )  e.  i )
4241anass1rs 782 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
z ( 2nd `  R
) x )  e.  i )
4342ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
4443an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( z ( 2nd `  R ) x )  e.  i ) )
453, 44sylan2 460 . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 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 2634 . . . . . . . . . . . 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 418 . . . . . . . . . . 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 5899 . . . . . . . . . . . 12  |-  ( z ( 2nd `  R
) x )  e. 
_V
5150elint2 3885 . . . . . . . . . . 11  |-  ( ( z ( 2nd `  R
) x )  e. 
|^| C  <->  A. i  e.  C  ( z
( 2nd `  R
) x )  e.  i )
5249, 51sylibr 203 . . . . . . . . . 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 26749 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( x  e.  i  /\  z  e.  ran  ( 1st `  R ) ) )  ->  (
x ( 2nd `  R
) z )  e.  i )
5453anass1rs 782 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e. 
ran  ( 1st `  R
) )  /\  x  e.  i )  ->  (
x ( 2nd `  R
) z )  e.  i )
5554ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
5655an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R ) )  /\  i  e.  ( Idl `  R ) )  -> 
( x  e.  i  ->  ( x ( 2nd `  R ) z )  e.  i ) )
573, 56sylan2 460 . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 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 2634 . . . . . . . . . . . 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 418 . . . . . . . . . . 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 5899 . . . . . . . . . . . 12  |-  ( x ( 2nd `  R
) z )  e. 
_V
6362elint2 3885 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  R
) z )  e. 
|^| C  <->  A. i  e.  C  ( x
( 2nd `  R
) z )  e.  i )
6461, 63sylibr 203 . . . . . . . . . 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 518 . . . . . . . . 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 779 . . . . . . . 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 2639 . . . . . . 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 518 . . . . . 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 423 . . . . 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 208 . . . 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 2638 . . 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 974 . 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 26742 . . 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 976 . 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 1135 1  |-  ( ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) )  ->  |^| C  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  inidl  26758  igenidl  26791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-idl 26738
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