Theorem gasubg 19082

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 19073	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2		gasubg.1	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	subggrp 18931	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	anim12ci 614	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	gaf 19075	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
7	6	adantr 481	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
8		simpr 485	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
9	5	subgss 18929	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	8, 9	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11		xpss1 5652	. . . . . 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 6709	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
14	2	subgbas 18932	. . . . . . 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 5662	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
17	16	feq2d 6654	. . . 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 481	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ (SubGrp‘𝐺))
20		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	20	subg0cl 18936	. . . . . . . 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 485	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
24		ovres 7520	. . . . . . 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 18934	. . . . . . . 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 7372	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
29	20	gagrpid 19074	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
30	29	adantlr 713	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
31	25, 28, 30	3eqtr3d 2784	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32		eqimss2 4001	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → (Base‘𝐻) ⊆ 𝑆)
35	34	sselda 3944	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ 𝑆)
36	34	sselda 3944	. . . . . . . 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 769	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
39	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
40	9	ad3antlr 729	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41		simprr 771	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
42	40, 41	sseldd 3945	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ (Base‘𝐺))
43	23	adantr 481	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥 ∈ 𝑌)
44	39, 42, 43	fovcdmd 7526	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 ⊕ 𝑥) ∈ 𝑌)
45		ovres 7520	. . . . . . . . . 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 7520	. . . . . . . . . . 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 7373	. . . . . . . . 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 773	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
51	40, 38	sseldd 3945	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐺))
52		eqid 2736	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	5, 52	gaass 19077	. . . . . . . . . 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 2786	. . . . . . . 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 18938	. . . . . . . . . . 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 7520	. . . . . . . . 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 17171	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
62	61	ad3antlr 729	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (+g‘𝐺) = (+g‘𝐻))
63	62	oveqd 7374	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐻)𝑧))
64	63	oveq1d 7372	. . . . . . . 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 2783	. . . . . . 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 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 512	. . . 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 512	. 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 2736	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
72		eqid 2736	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
73		eqid 2736	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
74	71, 72, 73	isga 19071	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ⊆ wss 3910   × cxp 5631   ↾ cres 5635  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083   ↾_s cress 17112  +_gcplusg 17133  0_gc0g 17321  Grpcgrp 18748  SubGrpcsubg 18922   GrpAct cga 19069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-subg 18925  df-ga 19070

This theorem is referenced by:  sylow3lem5  19413