Theorem gasubg 19253

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 19244	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2		gasubg.1	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	subggrp 19084	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	anim12ci 613	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	gaf 19246	. . . . . 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 19082	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	8, 9	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11		xpss1 5697	. . . . . 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 6764	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
14	2	subgbas 19085	. . . . . . 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 5707	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
17	16	feq2d 6708	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ⊕ ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
18	13, 17	mpbid 231	. . 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 2728	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	20	subg0cl 19089	. . . . . . . 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 7587	. . . . . . 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 19087	. . . . . . . 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 7435	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
29	20	gagrpid 19245	. . . . . . 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 2776	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32		eqimss2 4039	. . . . . . . . . . 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 3980	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ 𝑆)
36	34	sselda 3980	. . . . . . . 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 3981	. . . . . . . . . . 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 7593	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 ⊕ 𝑥) ∈ 𝑌)
45		ovres 7587	. . . . . . . . . 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 7587	. . . . . . . . . . 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 7436	. . . . . . . . 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 3981	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐺))
52		eqid 2728	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	5, 52	gaass 19248	. . . . . . . . . 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 1370	. . . . . . . . 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 2778	. . . . . . . 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 19091	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
57	56	3expb 1118	. . . . . . . . . 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 7587	. . . . . . . . 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 17271	. . . . . . . . . . 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 7437	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐻)𝑧))
64	63	oveq1d 7435	. . . . . . . 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 2775	. . . . . . 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 3197	. . . . 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 3143	. . 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 2728	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
72		eqid 2728	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
73		eqid 2728	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
74	71, 72, 73	isga 19242	. 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 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471   ⊆ wss 3947   × cxp 5676   ↾ cres 5680  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420  Basecbs 17180   ↾_s cress 17209  +_gcplusg 17233  0_gc0g 17421  Grpcgrp 18890  SubGrpcsubg 19075   GrpAct cga 19240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-subg 19078  df-ga 19241

This theorem is referenced by:  sylow3lem5  19586