Theorem subgintm 13784

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 3950	. . . 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 3222	. . . . . . 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 2231	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 13760	. . . . . 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 2605	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 3923	. . . 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 3237	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ ( Base ` G ) )
12		eqid 2231	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 13768	. . . . . 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 2605	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S ( 0g ` G ) e. g )
16		ssel 3221	. . . . . . . 8 |- ( S C_ (SubGrp ` G ) -> ( w e. S -> w e. (SubGrp ` G ) ) )
17	16	eximdv 1928	. . . . . . 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 13765	. . . . . . 7 |- ( w e. (SubGrp ` G ) -> G e. Grp )
20	19	exlimiv 1646	. . . . . 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 13611	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
23		elintg 3936	. . . . 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 2819	. . 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 531	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
30		elinti 3937	. . . . . . . . . . 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 533	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
34		elinti 3937	. . . . . . . . . . 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 2231	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
38	37	subgcl 13770	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
39	28, 32, 36, 38	syl3anc 1273	. . . . . . . 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 2605	. . . . . . 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 2805	. . . . . . . . . . 11 |- x e. _V
42	41	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> x e. _V )
43		plusgslid 13194	. . . . . . . . . . . 12 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13108	. . . . . . . . . . 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 2805	. . . . . . . . . . 11 |- y e. _V
47	46	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> y e. _V )
48		ovexg 6051	. . . . . . . . . 10 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
49	42, 45, 47, 48	syl3anc 1273	. . . . . . . . 9 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( x ( +g ` G ) y ) e. _V )
50		elintg 3936	. . . . . . . . 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 2605	. . . 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 2231	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
59	58	subginvcl 13769	. . . . . . 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 2605	. . . . 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 3227	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> x e. ( Base ` G ) )
64	5, 58	grpinvcl 13630	. . . . . . 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 3936	. . . . . 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 2605	. 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 13775	. . 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 1206	1 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   E.wex 1540   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   U.cuni 3893   |^|cint 3928   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583  SubGrpcsubg 13753

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756

This theorem is referenced by:  subrngintm  14225  subrgintm  14256