Theorem intidl 33828

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.

Step	Hyp	Ref	Expression
1		intssuni 4499	. . . 4 |- ( C =/= (/) -> |^| C C_ U. C )
2	1	3ad2ant2 1083	. . 3 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> |^| C C_ U. C )
3		ssel2 3598	. . . . . . . 8 |- ( ( C C_ ( Idl ` R ) /\ i e. C ) -> i e. ( Idl ` R ) )
4		eqid 2622	. . . . . . . . 9 |- ( 1st ` R ) = ( 1st ` R )
5		eqid 2622	. . . . . . . . 9 |- ran ( 1st ` R ) = ran ( 1st ` R )
6	4, 5	idlss 33815	. . . . . . . 8 |- ( ( R e. RingOps /\ i e. ( Idl ` R ) ) -> i C_ ran ( 1st ` R ) )
7	3, 6	sylan2 491	. . . . . . 7 |- ( ( R e. RingOps /\ ( C C_ ( Idl ` R ) /\ i e. C ) ) -> i C_ ran ( 1st ` R ) )
8	7	anassrs 680	. . . . . 6 |- ( ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) /\ i e. C ) -> i C_ ran ( 1st ` R ) )
9	8	ralrimiva 2966	. . . . 5 |- ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) -> A. i e. C i C_ ran ( 1st ` R ) )
10	9	3adant2 1080	. . . 4 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> A. i e. C i C_ ran ( 1st ` R ) )
11		unissb 4469	. . . 4 |- ( U. C C_ ran ( 1st ` R ) <-> A. i e. C i C_ ran ( 1st ` R ) )
12	10, 11	sylibr 224	. . 3 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> U. C C_ ran ( 1st ` R ) )
13	2, 12	sstrd 3613	. 2 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> |^| C C_ ran ( 1st ` R ) )
14		eqid 2622	. . . . . . . 8 |- (GId ` ( 1st ` R ) ) = (GId ` ( 1st ` R ) )
15	4, 14	idl0cl 33817	. . . . . . 7 |- ( ( R e. RingOps /\ i e. ( Idl ` R ) ) -> (GId ` ( 1st ` R ) ) e. i )
16	3, 15	sylan2 491	. . . . . 6 |- ( ( R e. RingOps /\ ( C C_ ( Idl ` R ) /\ i e. C ) ) -> (GId ` ( 1st ` R ) ) e. i )
17	16	anassrs 680	. . . . 5 |- ( ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) /\ i e. C ) -> (GId ` ( 1st ` R ) ) e. i )
18	17	ralrimiva 2966	. . . 4 |- ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) -> A. i e. C (GId ` ( 1st ` R ) ) e. i )
19		fvex 6201	. . . . 5 |- (GId ` ( 1st ` R ) ) e. _V
20	19	elint2 4482	. . . 4 |- ( (GId ` ( 1st ` R ) ) e. |^| C <-> A. i e. C (GId ` ( 1st ` R ) ) e. i )
21	18, 20	sylibr 224	. . 3 |- ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) -> (GId ` ( 1st ` R ) ) e. |^| C )
22	21	3adant2 1080	. 2 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> (GId ` ( 1st ` R ) ) e. |^| C )
23		vex 3203	. . . . . 6 |- x e. _V
24	23	elint2 4482	. . . . 5 |- ( x e. |^| C <-> A. i e. C x e. i )
25		vex 3203	. . . . . . . . . 10 |- y e. _V
26	25	elint2 4482	. . . . . . . . 9 |- ( y e. |^| C <-> A. i e. C y e. i )
27		r19.26 3064	. . . . . . . . . . 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 ) )
28	4	idladdcl 33818	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ ( x e. i /\ y e. i ) ) -> ( x ( 1st ` R ) y ) e. i )
29	28	ex 450	. . . . . . . . . . . . . . 15 |- ( ( R e. RingOps /\ i e. ( Idl ` R ) ) -> ( ( x e. i /\ y e. i ) -> ( x ( 1st ` R ) y ) e. i ) )
30	3, 29	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( R e. RingOps /\ ( C C_ ( Idl ` R ) /\ i e. C ) ) -> ( ( x e. i /\ y e. i ) -> ( x ( 1st ` R ) y ) e. i ) )
31	30	anassrs 680	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) /\ i e. C ) -> ( ( x e. i /\ y e. i ) -> ( x ( 1st ` R ) y ) e. i ) )
32	31	ralimdva 2962	. . . . . . . . . . . 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 6678	. . . . . . . . . . . . 13 |- ( x ( 1st ` R ) y ) e. _V
34	33	elint2 4482	. . . . . . . . . . . 12 |- ( ( x ( 1st ` R ) y ) e. |^| C <-> A. i e. C ( x ( 1st ` R ) y ) e. i )
35	32, 34	syl6ibr 242	. . . . . . . . . . 11 |- ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) -> ( A. i e. C ( x e. i /\ y e. i ) -> ( x ( 1st ` R ) y ) e. |^| C ) )
36	27, 35	syl5bir 233	. . . . . . . . . 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 ) )
37	36	expdimp 453	. . . . . . . . 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 ) )
38	26, 37	syl5bi 232	. . . . . . . 8 |- ( ( ( R e. RingOps /\ C C_ ( Idl ` R ) ) /\ A. i e. C x e. i ) -> ( y e. |^| C -> ( x ( 1st ` R ) y ) e. |^| C ) )
39	38	ralrimiv 2965	. . . . . . 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 2622	. . . . . . . . . . . . . . . . . . . 20 |- ( 2nd ` R ) = ( 2nd ` R )
41	4, 40, 5	idllmulcl 33819	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ ( x e. i /\ z e. ran ( 1st ` R ) ) ) -> ( z ( 2nd ` R ) x ) e. i )
42	41	anass1rs 849	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ z e. ran ( 1st ` R ) ) /\ x e. i ) -> ( z ( 2nd ` R ) x ) e. i )
43	42	ex 450	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( x e. i -> ( z ( 2nd ` R ) x ) e. i ) )
44	43	an32s 846	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RingOps /\ z e. ran ( 1st ` R ) ) /\ i e. ( Idl ` R ) ) -> ( x e. i -> ( z ( 2nd ` R ) x ) e. i ) )
45	3, 44	sylan2 491	. . . . . . . . . . . . . . 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 ) )
46	45	an4s 869	. . . . . . . . . . . . . 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 ) )
47	46	anassrs 680	. . . . . . . . . . . . 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 ) )
48	47	ralimdva 2962	. . . . . . . . . . . 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 ) )
49	48	imp 445	. . . . . . . . . . 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 6678	. . . . . . . . . . . 12 |- ( z ( 2nd ` R ) x ) e. _V
51	50	elint2 4482	. . . . . . . . . . 11 |- ( ( z ( 2nd ` R ) x ) e. |^| C <-> A. i e. C ( z ( 2nd ` R ) x ) e. i )
52	49, 51	sylibr 224	. . . . . . . . . 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 )
53	4, 40, 5	idlrmulcl 33820	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ ( x e. i /\ z e. ran ( 1st ` R ) ) ) -> ( x ( 2nd ` R ) z ) e. i )
54	53	anass1rs 849	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ z e. ran ( 1st ` R ) ) /\ x e. i ) -> ( x ( 2nd ` R ) z ) e. i )
55	54	ex 450	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. RingOps /\ i e. ( Idl ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( x e. i -> ( x ( 2nd ` R ) z ) e. i ) )
56	55	an32s 846	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RingOps /\ z e. ran ( 1st ` R ) ) /\ i e. ( Idl ` R ) ) -> ( x e. i -> ( x ( 2nd ` R ) z ) e. i ) )
57	3, 56	sylan2 491	. . . . . . . . . . . . . . 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 ) )
58	57	an4s 869	. . . . . . . . . . . . . 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 ) )
59	58	anassrs 680	. . . . . . . . . . . . 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 ) )
60	59	ralimdva 2962	. . . . . . . . . . . 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 ) )
61	60	imp 445	. . . . . . . . . . 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 6678	. . . . . . . . . . . 12 |- ( x ( 2nd ` R ) z ) e. _V
63	62	elint2 4482	. . . . . . . . . . 11 |- ( ( x ( 2nd ` R ) z ) e. |^| C <-> A. i e. C ( x ( 2nd ` R ) z ) e. i )
64	61, 63	sylibr 224	. . . . . . . . . 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 )
65	52, 64	jca 554	. . . . . . . . 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 ) )
66	65	an32s 846	. . . . . . . 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 ) )
67	66	ralrimiva 2966	. . . . . . 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 ) )
68	39, 67	jca 554	. . . . . 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 ) ) )
69	68	ex 450	. . . . 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 ) ) ) )
70	24, 69	syl5bi 232	. . . 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 ) ) ) )
71	70	ralrimiv 2965	. . 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 ) ) )
72	71	3adant2 1080	. 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 ) ) )
73	4, 40, 5, 14	isidl 33813	. . 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 ) ) ) ) )
74	73	3ad2ant1 1082	. 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 ) ) ) ) )
75	13, 22, 72, 74	mpbir3and 1245	1 |- ( ( R e. RingOps /\ C =/= (/) /\ C C_ ( Idl ` R ) ) -> |^| C e. ( Idl ` R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-idl 33809

This theorem is referenced by:  inidl  33829  igenidl  33862