Theorem subgga 17032

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 16898	. . 3 |- ( Y e. (SubGrp ` G ) -> H e. Grp )
3		subgga.1	. . . 4 |- X = ( Base ` G )
4		fvex 5889	. . . 4 |- ( Base ` G ) e. _V
5	3, 4	eqeltri 2545	. . 3 |- X e. _V
6	2, 5	jctir 547	. 2 |- ( Y e. (SubGrp ` G ) -> ( H e. Grp /\ X e. _V ) )
7		subgrcl 16900	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
8	7	adantr 472	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> G e. Grp )
9	3	subgss 16896	. . . . . . . . 9 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
10	9	sselda 3418	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. Y ) -> x e. X )
11	10	adantrr 731	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> x e. X )
12		simprr 774	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> y e. X )
13		subgga.2	. . . . . . . 8 |- .+ = ( +g ` G )
14	3, 13	grpcl 16757	. . . . . . 7 |- ( ( G e. Grp /\ x e. X /\ y e. X ) -> ( x .+ y ) e. X )
15	8, 11, 12, 14	syl3anc 1292	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ ( x e. Y /\ y e. X ) ) -> ( x .+ y ) e. X )
16	15	ralrimivva 2814	. . . . 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 6879	. . . . 5 |- ( A. x e. Y A. y e. X ( x .+ y ) e. X <-> F : ( Y X. X ) --> X )
19	16, 18	sylib 201	. . . 4 |- ( Y e. (SubGrp ` G ) -> F : ( Y X. X ) --> X )
20	1	subgbas 16899	. . . . . 6 |- ( Y e. (SubGrp ` G ) -> Y = ( Base ` H ) )
21	20	xpeq1d 4862	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( Y X. X ) = ( ( Base ` H ) X. X ) )
22	21	feq2d 5725	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( F : ( Y X. X ) --> X <-> F : ( ( Base ` H ) X. X ) --> X ) )
23	19, 22	mpbid 215	. . 3 |- ( Y e. (SubGrp ` G ) -> F : ( ( Base ` H ) X. X ) --> X )
24		eqid 2471	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
25	24	subg0cl 16903	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) e. Y )
26		oveq12 6317	. . . . . . . 8 |- ( ( x = ( 0g ` G ) /\ y = u ) -> ( x .+ y ) = ( ( 0g ` G ) .+ u ) )
27		ovex 6336	. . . . . . . 8 |- ( ( 0g ` G ) .+ u ) e. _V
28	26, 17, 27	ovmpt2a 6446	. . . . . . 7 |- ( ( ( 0g ` G ) e. Y /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
29	25, 28	sylan 479	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` G ) .+ u ) )
30	1, 24	subg0 16901	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
31	30	oveq1d 6323	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
32	31	adantr 472	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) F u ) = ( ( 0g ` H ) F u ) )
33	3, 13, 24	grplid 16774	. . . . . . 7 |- ( ( G e. Grp /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
34	7, 33	sylan 479	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` G ) .+ u ) = u )
35	29, 32, 34	3eqtr3d 2513	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ u e. X ) -> ( ( 0g ` H ) F u ) = u )
36	7	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> G e. Grp )
37	9	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> Y C_ X )
38		simprl 772	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. Y )
39	37, 38	sseldd 3419	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> v e. X )
40		simprr 774	. . . . . . . . . . 11 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. Y )
41	37, 40	sseldd 3419	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> w e. X )
42		simplr 770	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> u e. X )
43	3, 13	grpass 16758	. . . . . . . . . 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 1294	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v .+ ( w .+ u ) ) )
45	3, 13	grpcl 16757	. . . . . . . . . . 11 |- ( ( G e. Grp /\ w e. X /\ u e. X ) -> ( w .+ u ) e. X )
46	36, 41, 42, 45	syl3anc 1292	. . . . . . . . . 10 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w .+ u ) e. X )
47		oveq12 6317	. . . . . . . . . . 11 |- ( ( x = v /\ y = ( w .+ u ) ) -> ( x .+ y ) = ( v .+ ( w .+ u ) ) )
48		ovex 6336	. . . . . . . . . . 11 |- ( v .+ ( w .+ u ) ) e. _V
49	47, 17, 48	ovmpt2a 6446	. . . . . . . . . 10 |- ( ( v e. Y /\ ( w .+ u ) e. X ) -> ( v F ( w .+ u ) ) = ( v .+ ( w .+ u ) ) )
50	38, 46, 49	syl2anc 673	. . . . . . . . 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 2508	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) .+ u ) = ( v F ( w .+ u ) ) )
52	13	subgcl 16905	. . . . . . . . . . 11 |- ( ( Y e. (SubGrp ` G ) /\ v e. Y /\ w e. Y ) -> ( v .+ w ) e. Y )
53	52	3expb 1232	. . . . . . . . . 10 |- ( ( Y e. (SubGrp ` G ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
54	53	adantlr 729	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( v .+ w ) e. Y )
55		oveq12 6317	. . . . . . . . . 10 |- ( ( x = ( v .+ w ) /\ y = u ) -> ( x .+ y ) = ( ( v .+ w ) .+ u ) )
56		ovex 6336	. . . . . . . . . 10 |- ( ( v .+ w ) .+ u ) e. _V
57	55, 17, 56	ovmpt2a 6446	. . . . . . . . 9 |- ( ( ( v .+ w ) e. Y /\ u e. X ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
58	54, 42, 57	syl2anc 673	. . . . . . . 8 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( ( v .+ w ) F u ) = ( ( v .+ w ) .+ u ) )
59		oveq12 6317	. . . . . . . . . . 11 |- ( ( x = w /\ y = u ) -> ( x .+ y ) = ( w .+ u ) )
60		ovex 6336	. . . . . . . . . . 11 |- ( w .+ u ) e. _V
61	59, 17, 60	ovmpt2a 6446	. . . . . . . . . 10 |- ( ( w e. Y /\ u e. X ) -> ( w F u ) = ( w .+ u ) )
62	40, 42, 61	syl2anc 673	. . . . . . . . 9 |- ( ( ( Y e. (SubGrp ` G ) /\ u e. X ) /\ ( v e. Y /\ w e. Y ) ) -> ( w F u ) = ( w .+ u ) )
63	62	oveq2d 6324	. . . . . . . 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 2515	. . . . . . 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 2814	. . . . . 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 15317	. . . . . . . . . . . 12 |- ( Y e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
67	66	oveqd 6325	. . . . . . . . . . 11 |- ( Y e. (SubGrp ` G ) -> ( v .+ w ) = ( v ( +g ` H ) w ) )
68	67	oveq1d 6323	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> ( ( v .+ w ) F u ) = ( ( v ( +g ` H ) w ) F u ) )
69	68	eqeq1d 2473	. . . . . . . . 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 2987	. . . . . . . 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 2987	. . . . . . 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 492	. . . . . 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 478	. . . . 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 541	. . . 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 2809	. . 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 541	. 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 2471	. . 3 |- ( Base ` H ) = ( Base ` H )
78		eqid 2471	. . 3 |- ( +g ` H ) = ( +g ` H )
79		eqid 2471	. . 3 |- ( 0g ` H ) = ( 0g ` H )
80	77, 78, 79	isga 17023	. 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 677	1 |- ( Y e. (SubGrp ` G ) -> F e. ( H GrpAct X ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   C_ wss 3390   X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   Basecbs 15199   ↾_s cress 15200   +g cplusg 15268   0gc0g 15416   Grpcgrp 16747  SubGrpcsubg 16889   GrpAct cga 17021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-subg 16892  df-ga 17022

This theorem is referenced by:  gaid2  17035