Theorem subgintm 13268

Description: The intersection of an inhabited collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)

Assertion

Ref	Expression
subgintm	|- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Distinct variable groups:   w, G   w, S

Proof of Theorem subgintm

Dummy variables x g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		intssunim 3892	. . . 4 |- ( E. w w e. S -> |^| S C_ U. S )
2	1	adantl 277	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ U. S )
3		ssel2 3174	. . . . . . 7 |- ( ( S C_ (SubGrp ` G ) /\ g e. S ) -> g e. (SubGrp ` G ) )
4	3	adantlr 477	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ g e. S ) -> g e. (SubGrp ` G ) )
5		eqid 2193	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 13244	. . . . . 6 |- ( g e. (SubGrp ` G ) -> g C_ ( Base ` G ) )
7	4, 6	syl 14	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ g e. S ) -> g C_ ( Base ` G ) )
8	7	ralrimiva 2567	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 3865	. . . 4 |- ( U. S C_ ( Base ` G ) <-> A. g e. S g C_ ( Base ` G ) )
10	8, 9	sylibr 134	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> U. S C_ ( Base ` G ) )
11	2, 10	sstrd 3189	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ ( Base ` G ) )
12		eqid 2193	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 13252	. . . . . 6 |- ( g e. (SubGrp ` G ) -> ( 0g ` G ) e. g )
14	4, 13	syl 14	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ g e. S ) -> ( 0g ` G ) e. g )
15	14	ralrimiva 2567	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S ( 0g ` G ) e. g )
16		ssel 3173	. . . . . . . 8 |- ( S C_ (SubGrp ` G ) -> ( w e. S -> w e. (SubGrp ` G ) ) )
17	16	eximdv 1891	. . . . . . 7 |- ( S C_ (SubGrp ` G ) -> ( E. w w e. S -> E. w w e. (SubGrp ` G ) ) )
18	17	imp 124	. . . . . 6 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> E. w w e. (SubGrp ` G ) )
19		subgrcl 13249	. . . . . . 7 |- ( w e. (SubGrp ` G ) -> G e. Grp )
20	19	exlimiv 1609	. . . . . 6 |- ( E. w w e. (SubGrp ` G ) -> G e. Grp )
21	18, 20	syl 14	. . . . 5 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> G e. Grp )
22	5, 12	grpidcl 13101	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
23		elintg 3878	. . . . 5 |- ( ( 0g ` G ) e. ( Base ` G ) -> ( ( 0g ` G ) e. |^| S <-> A. g e. S ( 0g ` G ) e. g ) )
24	21, 22, 23	3syl 17	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( ( 0g ` G ) e. |^| S <-> A. g e. S ( 0g ` G ) e. g ) )
25	15, 24	mpbird 167	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( 0g ` G ) e. |^| S )
26		elex2 2776	. . 3 |- ( ( 0g ` G ) e. |^| S -> E. w w e. |^| S )
27	25, 26	syl 14	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> E. w w e. |^| S )
28	4	adantlr 477	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> g e. (SubGrp ` G ) )
29		simprl 529	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
30		elinti 3879	. . . . . . . . . . 11 |- ( x e. |^| S -> ( g e. S -> x e. g ) )
31	30	imp 124	. . . . . . . . . 10 |- ( ( x e. |^| S /\ g e. S ) -> x e. g )
32	29, 31	sylan 283	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> x e. g )
33		simprr 531	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
34		elinti 3879	. . . . . . . . . . 11 |- ( y e. |^| S -> ( g e. S -> y e. g ) )
35	34	imp 124	. . . . . . . . . 10 |- ( ( y e. |^| S /\ g e. S ) -> y e. g )
36	33, 35	sylan 283	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> y e. g )
37		eqid 2193	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
38	37	subgcl 13254	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
39	28, 32, 36, 38	syl3anc 1249	. . . . . . . 8 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> ( x ( +g ` G ) y ) e. g )
40	39	ralrimiva 2567	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. g e. S ( x ( +g ` G ) y ) e. g )
41		vex 2763	. . . . . . . . . . 11 |- x e. _V
42	41	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> x e. _V )
43		plusgslid 12730	. . . . . . . . . . . 12 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 12645	. . . . . . . . . . 11 |- ( G e. Grp -> ( +g ` G ) e. _V )
45	18, 20, 44	3syl 17	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( +g ` G ) e. _V )
46		vex 2763	. . . . . . . . . . 11 |- y e. _V
47	46	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> y e. _V )
48		ovexg 5952	. . . . . . . . . 10 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
49	42, 45, 47, 48	syl3anc 1249	. . . . . . . . 9 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( x ( +g ` G ) y ) e. _V )
50		elintg 3878	. . . . . . . . 9 |- ( ( x ( +g ` G ) y ) e. _V -> ( ( x ( +g ` G ) y ) e. |^| S <-> A. g e. S ( x ( +g ` G ) y ) e. g ) )
51	49, 50	syl 14	. . . . . . . 8 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( ( x ( +g ` G ) y ) e. |^| S <-> A. g e. S ( x ( +g ` G ) y ) e. g ) )
52	51	adantr 276	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( ( x ( +g ` G ) y ) e. |^| S <-> A. g e. S ( x ( +g ` G ) y ) e. g ) )
53	40, 52	mpbird 167	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( +g ` G ) y ) e. |^| S )
54	53	anassrs 400	. . . . 5 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ y e. |^| S ) -> ( x ( +g ` G ) y ) e. |^| S )
55	54	ralrimiva 2567	. . . 4 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S )
56	4	adantlr 477	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> g e. (SubGrp ` G ) )
57	31	adantll 476	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> x e. g )
58		eqid 2193	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
59	58	subginvcl 13253	. . . . . . 7 |- ( ( g e. (SubGrp ` G ) /\ x e. g ) -> ( ( invg ` G ) ` x ) e. g )
60	56, 57, 59	syl2anc 411	. . . . . 6 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> ( ( invg ` G ) ` x ) e. g )
61	60	ralrimiva 2567	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> A. g e. S ( ( invg ` G ) ` x ) e. g )
62	21	adantr 276	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> G e. Grp )
63	11	sselda 3179	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> x e. ( Base ` G ) )
64	5, 58	grpinvcl 13120	. . . . . . 7 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) )
65	62, 63, 64	syl2anc 411	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) )
66		elintg 3878	. . . . . 6 |- ( ( ( invg ` G ) ` x ) e. ( Base ` G ) -> ( ( ( invg ` G ) ` x ) e. |^| S <-> A. g e. S ( ( invg ` G ) ` x ) e. g ) )
67	65, 66	syl 14	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> ( ( ( invg ` G ) ` x ) e. |^| S <-> A. g e. S ( ( invg ` G ) ` x ) e. g ) )
68	61, 67	mpbird 167	. . . 4 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> ( ( invg ` G ) ` x ) e. |^| S )
69	55, 68	jca 306	. . 3 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) )
70	69	ralrimiva 2567	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) )
71	5, 37, 58	issubg2m 13259	. . 3 |- ( G e. Grp -> ( |^| S e. (SubGrp ` G ) <-> ( |^| S C_ ( Base ` G ) /\ E. w w e. |^| S /\ A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) ) ) )
72	18, 20, 71	3syl 17	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( |^| S e. (SubGrp ` G ) <-> ( |^| S C_ ( Base ` G ) /\ E. w w e. |^| S /\ A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) ) ) )
73	11, 27, 70, 72	mpbir3and 1182	1 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   E.wex 1503   e. wcel 2164   A.wral 2472   _Vcvv 2760   C_ wss 3153   U.cuni 3835   |^|cint 3870   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073  SubGrpcsubg 13237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240

This theorem is referenced by:  subrngintm  13708  subrgintm  13739