Theorem subgintm 13735

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 3945	. . . 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 3219	. . . . . . 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 2229	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 13711	. . . . . 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 2603	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 3918	. . . 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 3234	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ ( Base ` G ) )
12		eqid 2229	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 13719	. . . . . 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 2603	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S ( 0g ` G ) e. g )
16		ssel 3218	. . . . . . . 8 |- ( S C_ (SubGrp ` G ) -> ( w e. S -> w e. (SubGrp ` G ) ) )
17	16	eximdv 1926	. . . . . . 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 13716	. . . . . . 7 |- ( w e. (SubGrp ` G ) -> G e. Grp )
20	19	exlimiv 1644	. . . . . 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 13562	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
23		elintg 3931	. . . . 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 2816	. . 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 3932	. . . . . . . . . . 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 3932	. . . . . . . . . . 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 2229	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
38	37	subgcl 13721	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
39	28, 32, 36, 38	syl3anc 1271	. . . . . . . 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 2603	. . . . . . 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 2802	. . . . . . . . . . 11 |- x e. _V
42	41	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> x e. _V )
43		plusgslid 13145	. . . . . . . . . . . 12 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13059	. . . . . . . . . . 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 2802	. . . . . . . . . . 11 |- y e. _V
47	46	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> y e. _V )
48		ovexg 6035	. . . . . . . . . 10 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
49	42, 45, 47, 48	syl3anc 1271	. . . . . . . . 9 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( x ( +g ` G ) y ) e. _V )
50		elintg 3931	. . . . . . . . 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 2603	. . . 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 2229	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
59	58	subginvcl 13720	. . . . . . 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 2603	. . . . 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 3224	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> x e. ( Base ` G ) )
64	5, 58	grpinvcl 13581	. . . . . . 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 3931	. . . . . 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 2603	. 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 13726	. . 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 1204	1 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   U.cuni 3888   |^|cint 3923   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533   invgcminusg 13534  SubGrpcsubg 13704

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-subg 13707

This theorem is referenced by:  subrngintm  14176  subrgintm  14207