Theorem subgga 16212

Description: A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
subgga.1	|- X = ( Base ` G )
subgga.2	|- .+ = ( +g ` G )
subgga.3	|- H = ( G ↾s Y )
subgga.4	|- F = ( x e. Y , y e. X |-> ( x .+ y ) )

Assertion

Ref	Expression
subgga	|- ( Y e. (SubGrp ` G ) -> F e. ( H GrpAct X ) )

Distinct variable groups:   x, y, G   x, X, y   x, Y, y   x, .+ , y

Allowed substitution hints:   F( x, y)   H( x, y)

Proof of Theorem subgga

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgga.3	. . . 4 |- H = ( G ↾s Y )
2	1	subggrp 16078	. . 3 |- ( Y e. (SubGrp ` G ) -> H e. Grp )
3		subgga.1	. . . 4 |- X = ( Base ` G )
4		fvex 5866	. . . 4 |- ( Base ` G ) e. _V
5	3, 4	eqeltri 2527	. . 3 |- X e. _V
6	2, 5	jctir 538	. 2 |- ( Y e. (SubGrp ` G ) -> ( H e. Grp /\ X e. _V ) )
7		subgrcl 16080	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
8	7	adantr 465	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> G e. Grp )
9	3	subgss 16076	. . . . . . . . 9 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
10	9	sselda 3489	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. Y ) -> x e. X )
11	10	adantrr 716	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> x e. X )
12		simprr 757	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> y e. X )
13		subgga.2	. . . . . . . 8 |- .+ = ( +g ` G )
14	3, 13	grpcl 15937	. . . . . . 7 |- ( ( G e. Grp /\ x e. X /\ y e. X ) -> ( x .+ y ) e. X )
15	8, 11, 12, 14	syl3anc 1229	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> ( x .+ y ) e. X )
16	15	ralrimivva 2864	. . . . 5 |- ( Y e. (SubGrp ` G ) -> A. x e. Y A. y e. X ( x .+ y ) e. X )
17		subgga.4	. . . . . 6 |- F = ( x e. Y , y e. X |-> ( x .+ y ) )
18	17	fmpt2 6852	. . . . 5 |- ( A. x e. Y A. y e. X ( x .+ y ) e. X <-> F : ( Y X. X ) --> X )
19	16, 18	sylib 196	. . . 4 |- ( Y e. (SubGrp ` G ) -> F : ( Y X. X ) --> X )
20	1	subgbas 16079	. . . . . 6 |- ( Y e. (SubGrp ` G ) -> Y = ( Base ` H ) )
21	20	xpeq1d 5012	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( Y X. X ) = ( ( Base ` H ) X. X ) )
22	21	feq2d 5708	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( F : ( Y X. X ) --> X <-> F : ( ( Base ` H ) X. X ) --> X ) )
23	19, 22	mpbid 210	. . 3 |- ( Y e. (SubGrp ` G ) -> F : ( ( Base ` H ) X. X ) --> X )
24		eqid 2443	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
25	24	subg0cl 16083	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) e. Y )
26		oveq12 6290	. . . . . . . 8 |- ( ( x = ( 0g ` G ) /\ y = u ) -> ( x .+ y ) = ( ( 0g ` G ) .+ u ) )
27		ovex 6309	. . . . . . . 8 |- ( ( 0g ` G ) .+ u ) e. _V
28	26, 17, 27	ovmpt2a 6418	. . . . . . 7 |- ( ( ( 0g ` G ) e. Y /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
29	25, 28	sylan 471	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
30	1, 24	subg0 16081	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
31	30	oveq1d 6296	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
32	31	adantr 465	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
33	3, 13, 24	grplid 15954	. . . . . . 7 |- ( ( G e. Grp /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
34	7, 33	sylan 471	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
35	29, 32, 34	3eqtr3d 2492	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` H ) F u ) = u )
36	7	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> G e. Grp )
37	9	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> Y C_ X )
38		simprl 756	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. Y )
39	37, 38	sseldd 3490	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. X )
40		simprr 757	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. Y )
41	37, 40	sseldd 3490	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. X )
42		simplr 755	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> u e. X )
43	3, 13	grpass 15938	. . . . . . . . . 10 |- ( ( G e. Grp /\ ( v e. X /\ w e. X /\ u e. X ) ) -> ( ( v .+ w ) .+ u ) = ( v .+ ( w .+ u ) ) )
44	36, 39, 41, 42, 43	syl13anc 1231	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v .+ ( w .+ u ) ) )
45	3, 13	grpcl 15937	. . . . . . . . . . 11 |- ( ( G e. Grp /\ w e. X /\ u e. X ) -> ( w .+ u ) e. X )
46	36, 41, 42, 45	syl3anc 1229	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w .+ u ) e. X )
47		oveq12 6290	. . . . . . . . . . 11 |- ( ( x = v /\ y = ( w .+ u ) ) -> ( x .+ y ) = ( v .+ ( w .+ u ) ) )
48		ovex 6309	. . . . . . . . . . 11 |- ( v .+ ( w .+ u ) ) e. _V
49	47, 17, 48	ovmpt2a 6418	. . . . . . . . . 10 |- ( ( v e. Y /\ ( w .+ u ) e. X ) -> ( v F ( w .+ u ) ) = ( v .+ ( w .+ u ) ) )
50	38, 46, 49	syl2anc 661	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( v F ( w .+ u ) ) = ( v .+ ( w .+ u ) ) )
51	44, 50	eqtr4d 2487	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v F ( w .+ u ) ) )
52	13	subgcl 16085	. . . . . . . . . . 11 |- ( ( Y e. (SubGrp ` G ) /\ v e. Y /\ w e. Y ) -> ( v .+ w ) e. Y )
53	52	3expb 1198	. . . . . . . . . 10 |- ( ( Y e. (SubGrp ` G ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
54	53	adantlr 714	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
55		oveq12 6290	. . . . . . . . . 10 |- ( ( x = ( v .+ w ) /\ y = u ) -> ( x .+ y ) = ( ( v .+ w ) .+ u ) )
56		ovex 6309	. . . . . . . . . 10 |- ( ( v .+ w ) .+ u ) e. _V
57	55, 17, 56	ovmpt2a 6418	. . . . . . . . 9 |- ( ( ( v .+ w ) e. Y /\ u e. X ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
58	54, 42, 57	syl2anc 661	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
59		oveq12 6290	. . . . . . . . . . 11 |- ( ( x = w /\ y = u ) -> ( x .+ y ) = ( w .+ u ) )
60		ovex 6309	. . . . . . . . . . 11 |- ( w .+ u ) e. _V
61	59, 17, 60	ovmpt2a 6418	. . . . . . . . . 10 |- ( ( w e. Y /\ u e. X ) -> ( w F u ) = ( w .+ u ) )
62	40, 42, 61	syl2anc 661	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w F u ) = ( w .+ u ) )
63	62	oveq2d 6297	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( v F ( w F u ) ) = ( v F ( w .+ u ) ) )
64	51, 58, 63	3eqtr4d 2494	. . . . . . 7 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) F u ) = ( v F ( w F u ) ) )
65	64	ralrimivva 2864	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> A. v e. Y A. w e. Y ( ( v .+ w ) F u ) = ( v F ( w F u ) ) )
66	1, 13	ressplusg 14616	. . . . . . . . . . . 12 |- ( Y e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
67	66	oveqd 6298	. . . . . . . . . . 11 |- ( Y e. (SubGrp ` G ) -> ( v .+ w ) = ( v ( +g ` H ) w ) )
68	67	oveq1d 6296	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> ( ( v .+ w ) F u ) = ( ( v ( +g ` H ) w ) F u ) )
69	68	eqeq1d 2445	. . . . . . . . 9 |- ( Y e. (SubGrp ` G ) -> ( ( ( v .+ w ) F u ) = ( v F ( w F u ) ) <-> ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) )
70	20, 69	raleqbidv 3054	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> ( A. w e. Y ( ( v .+ w ) F u ) = ( v F ( w F u ) ) <-> A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) )
71	20, 70	raleqbidv 3054	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( A. v e. Y A. w e. Y ( ( v .+ w ) F u ) = ( v F ( w F u ) ) <-> A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) )
72	71	biimpa 484	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ A. v e. Y A. w e. Y ( ( v .+ w ) F u ) = ( v F ( w F u ) ) ) -> A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) )
73	65, 72	syldan 470	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) )
74	35, 73	jca 532	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( ( 0g ` H ) F u ) = u /\ A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) )
75	74	ralrimiva 2857	. . 3 |- ( Y e. (SubGrp ` G ) -> A. u e. X ( ( ( 0g ` H ) F u ) = u /\ A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) )
76	23, 75	jca 532	. 2 |- ( Y e. (SubGrp ` G ) -> ( F : ( ( Base ` H ) X. X ) --> X /\ A. u e. X ( ( ( 0g ` H ) F u ) = u /\ A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) ) )
77		eqid 2443	. . 3 |- ( Base ` H ) = ( Base ` H )
78		eqid 2443	. . 3 |- ( +g ` H ) = ( +g ` H )
79		eqid 2443	. . 3 |- ( 0g ` H ) = ( 0g ` H )
80	77, 78, 79	isga 16203	. 2 |- ( F e. ( H GrpAct X ) <-> ( ( H e. Grp /\ X e. _V ) /\ ( F : ( ( Base ` H ) X. X ) --> X /\ A. u e. X ( ( ( 0g ` H ) F u ) = u /\ A. v e. ( Base ` H ) A. w e. ( Base ` H ) ( ( v ( +g ` H ) w ) F u ) = ( v F ( w F u ) ) ) ) ) )
81	6, 76, 80	sylanbrc 664	1 |- ( Y e. (SubGrp ` G ) -> F e. ( H GrpAct X ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1383   e. wcel 1804   A.wral 2793   _Vcvv 3095   C_ wss 3461   X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281   |-> cmpt2 6283   Basecbs 14509   ↾_s cress 14510   +g cplusg 14574   0gc0g 14714   Grpcgrp 15927  SubGrpcsubg 16069   GrpAct cga 16201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-subg 16072  df-ga 16202

This theorem is referenced by:  gaid2  16215