Theorem subgintm 13932

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 3973	. . . 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 3235	. . . . . . 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 2234	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 13908	. . . . . 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 2617	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S g C_ ( Base ` G ) )
9		unissb 3946	. . . 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 3250	. 2 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S C_ ( Base ` G ) )
12		eqid 2234	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
13	12	subg0cl 13916	. . . . . 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 2617	. . . 4 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> A. g e. S ( 0g ` G ) e. g )
16		ssel 3234	. . . . . . . 8 |- ( S C_ (SubGrp ` G ) -> ( w e. S -> w e. (SubGrp ` G ) ) )
17	16	eximdv 1929	. . . . . . 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 13913	. . . . . . 7 |- ( w e. (SubGrp ` G ) -> G e. Grp )
20	19	exlimiv 1647	. . . . . 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 13759	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
23		elintg 3959	. . . . 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 2832	. . 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 3960	. . . . . . . . . . 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 3960	. . . . . . . . . . 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 2234	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
38	37	subgcl 13918	. . . . . . . . 9 |- ( ( g e. (SubGrp ` G ) /\ x e. g /\ y e. g ) -> ( x ( +g ` G ) y ) e. g )
39	28, 32, 36, 38	syl3anc 1274	. . . . . . . 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 2617	. . . . . . 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 2818	. . . . . . . . . . 11 |- x e. _V
42	41	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> x e. _V )
43		plusgslid 13342	. . . . . . . . . . . 12 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13256	. . . . . . . . . . 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 2818	. . . . . . . . . . 11 |- y e. _V
47	46	a1i 9	. . . . . . . . . 10 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> y e. _V )
48		ovexg 6086	. . . . . . . . . 10 |- ( ( x e. _V /\ ( +g ` G ) e. _V /\ y e. _V ) -> ( x ( +g ` G ) y ) e. _V )
49	42, 45, 47, 48	syl3anc 1274	. . . . . . . . 9 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> ( x ( +g ` G ) y ) e. _V )
50		elintg 3959	. . . . . . . . 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 2617	. . . 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 2234	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
59	58	subginvcl 13917	. . . . . . 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 2617	. . . . 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 3240	. . . . . . 7 |- ( ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) /\ x e. |^| S ) -> x e. ( Base ` G ) )
64	5, 58	grpinvcl 13778	. . . . . . 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 3959	. . . . . 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 2617	. 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 13923	. . 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 1207	1 |- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   E.wex 1541   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3213   U.cuni 3916   |^|cint 3951   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   0gc0g 13486   Grpcgrp 13730   invgcminusg 13731  SubGrpcsubg 13901

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-subg 13904

This theorem is referenced by:  subrngintm  14374  subrgintm  14405