Theorem gasubg 19214

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 19205	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2		gasubg.1	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	subggrp 19042	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	anim12ci 614	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	gaf 19207	. . . . . 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 19040	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	8, 9	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11		xpss1 5633	. . . . . 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 6690	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
14	2	subgbas 19043	. . . . . . 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 5643	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
17	16	feq2d 6635	. . . 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 2731	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	20	subg0cl 19047	. . . . . . . 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 7512	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑆 ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐺) ⊕ 𝑥))
25	22, 23, 24	syl2anc 584	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐺) ⊕ 𝑥))
26	2, 20	subg0 19045	. . . . . . . 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 7361	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
29	20	gagrpid 19206	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
30	29	adantlr 715	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
31	25, 28, 30	3eqtr3d 2774	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32		eqimss2 3989	. . . . . . . . . . 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 3929	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ 𝑆)
36	34	sselda 3929	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ (Base‘𝐻)) → 𝑧 ∈ 𝑆)
37	35, 36	anim12dan 619	. . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
40	9	ad3antlr 731	. . . . . . . . . . . 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 3930	. . . . . . . . . . 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 7518	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 ⊕ 𝑥) ∈ 𝑌)
45		ovres 7512	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝑧 ⊕ 𝑥) ∈ 𝑌) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
46	38, 44, 45	syl2anc 584	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦( ⊕ ↾ (𝑆 × 𝑌))(𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
47		ovres 7512	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑥 ∈ 𝑌) → (𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑧 ⊕ 𝑥))
48	41, 43, 47	syl2anc 584	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧( ⊕ ↾ (𝑆 × 𝑌))𝑥) = (𝑧 ⊕ 𝑥))
49	48	oveq2d 7362	. . . . . . . . 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 3930	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐺))
52		eqid 2731	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	5, 52	gaass 19209	. . . . . . . . . 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 1374	. . . . . . . . 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 2776	. . . . . . . 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 19049	. . . . . . . . . . 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 580	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
59		ovres 7512	. . . . . . . . 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 584	. . . . . . . 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 17195	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
62	61	ad3antlr 731	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (+g‘𝐺) = (+g‘𝐻))
63	62	oveqd 7363	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐻)𝑧))
64	63	oveq1d 7361	. . . . . . . 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 2773	. . . . . . 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 591	. . . . . 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 3175	. . . . 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 3124	. . 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 2731	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
72		eqid 2731	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
73		eqid 2731	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
74	71, 72, 73	isga 19203	. 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 583	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   × cxp 5612   ↾ cres 5616  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120   ↾_s cress 17141  +_gcplusg 17161  0_gc0g 17343  Grpcgrp 18846  SubGrpcsubg 19033   GrpAct cga 19201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-0g 17345  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-subg 19036  df-ga 19202

This theorem is referenced by:  sylow3lem5  19543