Theorem gasubg 19342

Description: The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypothesis

Ref	Expression
gasubg.1	⊢ 𝐻 = (𝐺 ↾s 𝑆)

Assertion

Ref	Expression
gasubg	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Proof of Theorem gasubg

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

Step	Hyp	Ref	Expression
1		gaset 19333	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2		gasubg.1	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	subggrp 19169	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	anim12ci 613	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5		eqid 2740	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	gaf 19335	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
7	6	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
8		simpr 484	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
9	5	subgss 19167	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	8, 9	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11		xpss1 5719	. . . . . 6 ⊢ (𝑆 ⊆ (Base‘𝐺) → (𝑆 × 𝑌) ⊆ ((Base‘𝐺) × 𝑌))
12	10, 11	syl 17	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) ⊆ ((Base‘𝐺) × 𝑌))
13	7, 12	fssresd 6788	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
14	2	subgbas 19170	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
15	8, 14	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
16	15	xpeq1d 5729	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
17	16	feq2d 6733	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ⊕ ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
18	13, 17	mpbid 232	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌)
19	8	adantr 480	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ (SubGrp‘𝐺))
20		eqid 2740	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	20	subg0cl 19174	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
22	19, 21	syl 17	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑆)
23		simpr 484	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
24		ovres 7616	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑆 ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐺) ⊕ 𝑥))
25	22, 23, 24	syl2anc 583	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐺) ⊕ 𝑥))
26	2, 20	subg0 19172	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
27	19, 26	syl 17	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → (0g‘𝐺) = (0g‘𝐻))
28	27	oveq1d 7463	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
29	20	gagrpid 19334	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
30	29	adantlr 714	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
31	25, 28, 30	3eqtr3d 2788	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32		eqimss2 4068	. . . . . . . . . . 11 ⊢ (𝑆 = (Base‘𝐻) → (Base‘𝐻) ⊆ 𝑆)
33	15, 32	syl 17	. . . . . . . . . 10 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐻) ⊆ 𝑆)
34	33	adantr 480	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → (Base‘𝐻) ⊆ 𝑆)
35	34	sselda 4008	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ 𝑆)
36	34	sselda 4008	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ (Base‘𝐻)) → 𝑧 ∈ 𝑆)
37	35, 36	anim12dan 618	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆))
38		simprl 770	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
39	7	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
40	9	ad3antlr 730	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41		simprr 772	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
42	40, 41	sseldd 4009	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐺))
43	23	adantr 480	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥 ∈ 𝑌)
44	39, 42, 43	fovcdmd 7622	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 ⊕ 𝑥) ∈ 𝑌)
45		ovres 7616	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝑧 ⊕ 𝑥) ∈ 𝑌) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
46	38, 44, 45	syl2anc 583	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
47		ovres 7616	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑌) → (𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑧 ⊕ 𝑥))
48	41, 43, 47	syl2anc 583	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑧 ⊕ 𝑥))
49	48	oveq2d 7464	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧 ⊕ 𝑥)))
50		simplll 774	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
51	40, 38	sseldd 4009	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐺))
52		eqid 2740	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	5, 52	gaass 19337	. . . . . . . . . 10 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥 ∈ 𝑌)) → ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
54	50, 51, 42, 43, 53	syl13anc 1372	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
55	46, 49, 54	3eqtr4d 2790	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥))
56	52	subgcl 19176	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
57	56	3expb 1120	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
58	19, 57	sylan 579	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
59		ovres 7616	. . . . . . . . 9 ⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 ∧ 𝑥 ∈ 𝑌) → ((𝑦(+g‘𝐺)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥))
60	58, 43, 59	syl2anc 583	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦(+g‘𝐺)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥))
61	2, 52	ressplusg 17349	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
62	61	ad3antlr 730	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (+g‘𝐺) = (+g‘𝐻))
63	62	oveqd 7465	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐻)𝑧))
64	63	oveq1d 7463	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦(+g‘𝐺)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
65	55, 60, 64	3eqtr2rd 2787	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)))
66	37, 65	syldan 590	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → ((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)))
67	66	ralrimivva 3208	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)))
68	31, 67	jca 511	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → (((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥))))
69	68	ralrimiva 3152	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ 𝑌 (((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥))))
70	18, 69	jca 511	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ⊕ ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥)))))
71		eqid 2740	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
72		eqid 2740	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
73		eqid 2740	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
74	71, 72, 73	isga 19331	. 2 ⊢ (( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌) ↔ ((𝐻 ∈ Grp ∧ 𝑌 ∈ V) ∧ (( ⊕ ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑧)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥))))))
75	4, 70, 74	sylanbrc 582	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976   × cxp 5698   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973  SubGrpcsubg 19160   GrpAct cga 19329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-subg 19163  df-ga 19330

This theorem is referenced by:  sylow3lem5  19673