Theorem subgintm 13978

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 3987	. . . 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 3243	. . . . . . 7 |- ( ( S C_ (SubGrp ` G ) /\ g e. S ) -> g e. (SubGrp ` G ) )
4	3	adantlr 481	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ g e. S ) -> g e. (SubGrp ` G ) )
5		eqid 2238	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 13954	. . . . . 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 2623	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 3960	. . . 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 3258	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ ( Base ` G ) )
12		eqid 2238	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 13962	. . . . . 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 2623	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S ( 0g ` G ) e. g )
16		ssel 3242	. . . . . . . 8 |- ( S C_ (SubGrp ` G ) -> ( w e. S -> w e. (SubGrp ` G ) ) )
17	16	eximdv 1933	. . . . . . 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 13959	. . . . . . 7 |- ( w e. (SubGrp ` G ) -> G e. Grp )
20	19	exlimiv 1651	. . . . . 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 13811	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
23		elintg 3973	. . . . 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 2838	. . 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 481	. . . . . . . . 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 535	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
30		elinti 3974	. . . . . . . . . . 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 537	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
34		elinti 3974	. . . . . . . . . . 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 2238	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
38	37	subgcl 13964	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
39	28, 32, 36, 38	syl3anc 1278	. . . . . . . 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 2623	. . . . . . 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 2824	. . . . . . . . . . 11 |- x e. _V
42	41	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> x e. _V )
43		plusgslid 13443	. . . . . . . . . . . 12 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13357	. . . . . . . . . . 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 2824	. . . . . . . . . . 11 |- y e. _V
47	46	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> y e. _V )
48		ovexg 6109	. . . . . . . . . 10 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
49	42, 45, 47, 48	syl3anc 1278	. . . . . . . . 9 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( x ( +g ` G ) y ) e. _V )
50		elintg 3973	. . . . . . . . 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 404	. . . . 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 2623	. . . 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 481	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> g e. (SubGrp ` G ) )
57	31	adantll 480	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> x e. g )
58		eqid 2238	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
59	58	subginvcl 13963	. . . . . . 7 |- ( ( g e. (SubGrp ` G ) /\ x e. g ) -> ( ( invg ` G ) ` x ) e. g )
60	56, 57, 59	syl2anc 415	. . . . . 6 |- ( ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) /\ g e. S ) -> ( ( invg ` G ) ` x ) e. g )
61	60	ralrimiva 2623	. . . . 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 3248	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> x e. ( Base ` G ) )
64	5, 58	grpinvcl 13830	. . . . . . 7 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) )
65	62, 63, 64	syl2anc 415	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) )
66		elintg 3973	. . . . . 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 2623	. 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 13969	. . 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 1211	1 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   U.cuni 3930   |^|cint 3965   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782   invgcminusg 13783  SubGrpcsubg 13947

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950

This theorem is referenced by:  subrngintm  14493  subrgintm  14524