Theorem subgint 17618

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

Assertion

Ref	Expression
subgint	|- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> |^| S e. (SubGrp ` G ) )

Proof of Theorem subgint

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

Step	Hyp	Ref	Expression
1		intssuni 4499	. . . 4 |- ( S =/= (/) -> |^| S C_ U. S )
2	1	adantl 482	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> |^| S C_ U. S )
3		ssel2 3598	. . . . . . 7 |- ( ( S C_ (SubGrp ` G ) /\ g e. S ) -> g e. (SubGrp ` G ) )
4	3	adantlr 751	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ g e. S ) -> g e. (SubGrp ` G ) )
5		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 17595	. . . . . 6 |- ( g e. (SubGrp ` G ) -> g C_ ( Base ` G ) )
7	4, 6	syl 17	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ g e. S ) -> g C_ ( Base ` G ) )
8	7	ralrimiva 2966	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 4469	. . . 4 |- ( U. S C_ ( Base ` G ) <-> A. g e. S g C_ ( Base ` G ) )
10	8, 9	sylibr 224	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> U. S C_ ( Base ` G ) )
11	2, 10	sstrd 3613	. 2 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> |^| S C_ ( Base ` G ) )
12		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 17602	. . . . . 6 |- ( g e. (SubGrp ` G ) -> ( 0g ` G ) e. g )
14	4, 13	syl 17	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ g e. S ) -> ( 0g ` G ) e. g )
15	14	ralrimiva 2966	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> A. g e. S ( 0g ` G ) e. g )
16		fvex 6201	. . . . 5 |- ( 0g ` G ) e. _V
17	16	elint2 4482	. . . 4 |- ( ( 0g ` G ) e. |^| S <-> A. g e. S ( 0g ` G ) e. g )
18	15, 17	sylibr 224	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> ( 0g ` G ) e. |^| S )
19		ne0i 3921	. . 3 |- ( ( 0g ` G ) e. |^| S -> |^| S =/= (/) )
20	18, 19	syl 17	. 2 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> |^| S =/= (/) )
21	4	adantlr 751	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> g e. (SubGrp ` G ) )
22		simprl 794	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
23		elinti 4485	. . . . . . . . . . 11 |- ( x e. |^| S -> ( g e. S -> x e. g ) )
24	23	imp 445	. . . . . . . . . 10 |- ( ( x e. |^| S /\ g e. S ) -> x e. g )
25	22, 24	sylan 488	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> x e. g )
26		simprr 796	. . . . . . . . . 10 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
27		elinti 4485	. . . . . . . . . . 11 |- ( y e. |^| S -> ( g e. S -> y e. g ) )
28	27	imp 445	. . . . . . . . . 10 |- ( ( y e. |^| S /\ g e. S ) -> y e. g )
29	26, 28	sylan 488	. . . . . . . . 9 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> y e. g )
30		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
31	30	subgcl 17604	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
32	21, 25, 29, 31	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ g e. S ) -> ( x ( +g ` G ) y ) e. g )
33	32	ralrimiva 2966	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. g e. S ( x ( +g ` G ) y ) e. g )
34		ovex 6678	. . . . . . . 8 |- ( x ( +g ` G ) y ) e. _V
35	34	elint2 4482	. . . . . . 7 |- ( ( x ( +g ` G ) y ) e. |^| S <-> A. g e. S ( x ( +g ` G ) y ) e. g )
36	33, 35	sylibr 224	. . . . . 6 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( +g ` G ) y ) e. |^| S )
37	36	anassrs 680	. . . . 5 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) /\ y e. |^| S ) -> ( x ( +g ` G ) y ) e. |^| S )
38	37	ralrimiva 2966	. . . 4 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) -> A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S )
39	4	adantlr 751	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) /\ g e. S ) -> g e. (SubGrp ` G ) )
40	24	adantll 750	. . . . . . 7 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) /\ g e. S ) -> x e. g )
41		eqid 2622	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
42	41	subginvcl 17603	. . . . . . 7 |- ( ( g e. (SubGrp ` G ) /\ x e. g ) -> ( ( invg ` G ) ` x ) e. g )
43	39, 40, 42	syl2anc 693	. . . . . 6 |- ( ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) /\ g e. S ) -> ( ( invg ` G ) ` x ) e. g )
44	43	ralrimiva 2966	. . . . 5 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) -> A. g e. S ( ( invg ` G ) ` x ) e. g )
45		fvex 6201	. . . . . 6 |- ( ( invg ` G ) ` x ) e. _V
46	45	elint2 4482	. . . . 5 |- ( ( ( invg ` G ) ` x ) e. |^| S <-> A. g e. S ( ( invg ` G ) ` x ) e. g )
47	44, 46	sylibr 224	. . . 4 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) -> ( ( invg ` G ) ` x ) e. |^| S )
48	38, 47	jca 554	. . 3 |- ( ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) /\ x e. |^| S ) -> ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) )
49	48	ralrimiva 2966	. 2 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) )
50		ssn0 3976	. . 3 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> (SubGrp ` G ) =/= (/) )
51		n0 3931	. . . 4 |- ( (SubGrp ` G ) =/= (/) <-> E. g g e. (SubGrp ` G ) )
52		subgrcl 17599	. . . . 5 |- ( g e. (SubGrp ` G ) -> G e. Grp )
53	52	exlimiv 1858	. . . 4 |- ( E. g g e. (SubGrp ` G ) -> G e. Grp )
54	51, 53	sylbi 207	. . 3 |- ( (SubGrp ` G ) =/= (/) -> G e. Grp )
55	5, 30, 41	issubg2 17609	. . 3 |- ( G e. Grp -> ( |^| S e. (SubGrp ` G ) <-> ( |^| S C_ ( Base ` G ) /\ |^| S =/= (/) /\ A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) ) ) )
56	50, 54, 55	3syl 18	. 2 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> ( |^| S e. (SubGrp ` G ) <-> ( |^| S C_ ( Base ` G ) /\ |^| S =/= (/) /\ A. x e. |^| S ( A. y e. |^| S ( x ( +g ` G ) y ) e. |^| S /\ ( ( invg ` G ) ` x ) e. |^| S ) ) ) )
57	11, 20, 49, 56	mpbir3and 1245	1 |- ( ( S C_ (SubGrp ` G ) /\ S =/= (/) ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423  SubGrpcsubg 17588

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591

This theorem is referenced by:  subrgint  18802