Theorem subgga 17733

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 17597	. . 3 |- ( Y e. (SubGrp ` G ) -> H e. Grp )
3		subgga.1	. . . 4 |- X = ( Base ` G )
4		fvex 6201	. . . 4 |- ( Base ` G ) e. _V
5	3, 4	eqeltri 2697	. . 3 |- X e. _V
6	2, 5	jctir 561	. 2 |- ( Y e. (SubGrp ` G ) -> ( H e. Grp /\ X e. _V ) )
7		subgrcl 17599	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
8	7	adantr 481	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> G e. Grp )
9	3	subgss 17595	. . . . . . . . 9 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
10	9	sselda 3603	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. Y ) -> x e. X )
11	10	adantrr 753	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> x e. X )
12		simprr 796	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> y e. X )
13		subgga.2	. . . . . . . 8 |- .+ = ( +g ` G )
14	3, 13	grpcl 17430	. . . . . . 7 |- ( ( G e. Grp /\ x e. X /\ y e. X ) -> ( x .+ y ) e. X )
15	8, 11, 12, 14	syl3anc 1326	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> ( x .+ y ) e. X )
16	15	ralrimivva 2971	. . . . 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 7237	. . . . 5 |- ( A. x e. Y A. y e. X ( x .+ y ) e. X <-> F : ( Y X. X ) --> X )
19	16, 18	sylib 208	. . . 4 |- ( Y e. (SubGrp ` G ) -> F : ( Y X. X ) --> X )
20	1	subgbas 17598	. . . . . 6 |- ( Y e. (SubGrp ` G ) -> Y = ( Base ` H ) )
21	20	xpeq1d 5138	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( Y X. X ) = ( ( Base ` H ) X. X ) )
22	21	feq2d 6031	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( F : ( Y X. X ) --> X <-> F : ( ( Base ` H ) X. X ) --> X ) )
23	19, 22	mpbid 222	. . 3 |- ( Y e. (SubGrp ` G ) -> F : ( ( Base ` H ) X. X ) --> X )
24		eqid 2622	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
25	24	subg0cl 17602	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) e. Y )
26		oveq12 6659	. . . . . . . 8 |- ( ( x = ( 0g ` G ) /\ y = u ) -> ( x .+ y ) = ( ( 0g ` G ) .+ u ) )
27		ovex 6678	. . . . . . . 8 |- ( ( 0g ` G ) .+ u ) e. _V
28	26, 17, 27	ovmpt2a 6791	. . . . . . 7 |- ( ( ( 0g ` G ) e. Y /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
29	25, 28	sylan 488	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
30	1, 24	subg0 17600	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
31	30	oveq1d 6665	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
32	31	adantr 481	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
33	3, 13, 24	grplid 17452	. . . . . . 7 |- ( ( G e. Grp /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
34	7, 33	sylan 488	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
35	29, 32, 34	3eqtr3d 2664	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` H ) F u ) = u )
36	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> G e. Grp )
37	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> Y C_ X )
38		simprl 794	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. Y )
39	37, 38	sseldd 3604	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. X )
40		simprr 796	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. Y )
41	37, 40	sseldd 3604	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. X )
42		simplr 792	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> u e. X )
43	3, 13	grpass 17431	. . . . . . . . . 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 1328	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v .+ ( w .+ u ) ) )
45	3, 13	grpcl 17430	. . . . . . . . . . 11 |- ( ( G e. Grp /\ w e. X /\ u e. X ) -> ( w .+ u ) e. X )
46	36, 41, 42, 45	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w .+ u ) e. X )
47		oveq12 6659	. . . . . . . . . . 11 |- ( ( x = v /\ y = ( w .+ u ) ) -> ( x .+ y ) = ( v .+ ( w .+ u ) ) )
48		ovex 6678	. . . . . . . . . . 11 |- ( v .+ ( w .+ u ) ) e. _V
49	47, 17, 48	ovmpt2a 6791	. . . . . . . . . 10 |- ( ( v e. Y /\ ( w .+ u ) e. X ) -> ( v F ( w .+ u ) ) = ( v .+ ( w .+ u ) ) )
50	38, 46, 49	syl2anc 693	. . . . . . . . 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 2659	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v F ( w .+ u ) ) )
52	13	subgcl 17604	. . . . . . . . . . 11 |- ( ( Y e. (SubGrp ` G ) /\ v e. Y /\ w e. Y ) -> ( v .+ w ) e. Y )
53	52	3expb 1266	. . . . . . . . . 10 |- ( ( Y e. (SubGrp ` G ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
54	53	adantlr 751	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
55		oveq12 6659	. . . . . . . . . 10 |- ( ( x = ( v .+ w ) /\ y = u ) -> ( x .+ y ) = ( ( v .+ w ) .+ u ) )
56		ovex 6678	. . . . . . . . . 10 |- ( ( v .+ w ) .+ u ) e. _V
57	55, 17, 56	ovmpt2a 6791	. . . . . . . . 9 |- ( ( ( v .+ w ) e. Y /\ u e. X ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
58	54, 42, 57	syl2anc 693	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
59		oveq12 6659	. . . . . . . . . . 11 |- ( ( x = w /\ y = u ) -> ( x .+ y ) = ( w .+ u ) )
60		ovex 6678	. . . . . . . . . . 11 |- ( w .+ u ) e. _V
61	59, 17, 60	ovmpt2a 6791	. . . . . . . . . 10 |- ( ( w e. Y /\ u e. X ) -> ( w F u ) = ( w .+ u ) )
62	40, 42, 61	syl2anc 693	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w F u ) = ( w .+ u ) )
63	62	oveq2d 6666	. . . . . . . 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 2666	. . . . . . 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 2971	. . . . . 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 15993	. . . . . . . . . . . 12 |- ( Y e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
67	66	oveqd 6667	. . . . . . . . . . 11 |- ( Y e. (SubGrp ` G ) -> ( v .+ w ) = ( v ( +g ` H ) w ) )
68	67	oveq1d 6665	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> ( ( v .+ w ) F u ) = ( ( v ( +g ` H ) w ) F u ) )
69	68	eqeq1d 2624	. . . . . . . . 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 3152	. . . . . . . 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 3152	. . . . . . 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 501	. . . . . 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 487	. . . . 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 554	. . . 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 2966	. . 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 554	. 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 2622	. . 3 |- ( Base ` H ) = ( Base ` H )
78		eqid 2622	. . 3 |- ( +g ` H ) = ( +g ` H )
79		eqid 2622	. . 3 |- ( 0g ` H ) = ( 0g ` H )
80	77, 78, 79	isga 17724	. 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 698	1 |- ( Y e. (SubGrp ` G ) -> F e. ( H GrpAct X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   GrpAct cga 17722

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-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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  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-subg 17591  df-ga 17723

This theorem is referenced by:  gaid2  17736