Theorem gasubg 18908

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 18899	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2		gasubg.1	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	subggrp 18758	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
4	1, 3	anim12ci 614	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	gaf 18901	. . . . . 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 18756	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	8, 9	syl 17	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11		xpss1 5608	. . . . . 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 6641	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
14	2	subgbas 18759	. . . . . . 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 5618	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
17	16	feq2d 6586	. . . 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 2738	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	20	subg0cl 18763	. . . . . . . 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 7438	. . . . . . 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 18761	. . . . . . . 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 7290	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥))
29	20	gagrpid 18900	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
30	29	adantlr 712	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
31	25, 28, 30	3eqtr3d 2786	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐻)( ⊕ ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32		eqimss2 3978	. . . . . . . . . . 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 3921	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦 ∈ 𝑆)
36	34	sselda 3921	. . . . . . . 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 768	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
39	7	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌)
40	9	ad3antlr 728	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41		simprr 770	. . . . . . . . . . . 12 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
42	40, 41	sseldd 3922	. . . . . . . . . . 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	fovrnd 7444	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 ⊕ 𝑥) ∈ 𝑌)
45		ovres 7438	. . . . . . . . . 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 7438	. . . . . . . . . . 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 7291	. . . . . . . . 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 772	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
51	40, 38	sseldd 3922	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐺))
52		eqid 2738	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	5, 52	gaass 18903	. . . . . . . . . 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 1371	. . . . . . . . 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 2788	. . . . . . . 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 18765	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
57	56	3expb 1119	. . . . . . . . . 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 7438	. . . . . . . . 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 17000	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
62	61	ad3antlr 728	. . . . . . . . . 10 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (+g‘𝐺) = (+g‘𝐻))
63	62	oveqd 7292	. . . . . . . . 9 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑌) ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐻)𝑧))
64	63	oveq1d 7290	. . . . . . . 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 2785	. . . . . . 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 3123	. . . . 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 3103	. . 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 2738	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
72		eqid 2738	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
73		eqid 2738	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
74	71, 72, 73	isga 18897	. 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 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   × cxp 5587   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912   ↾_s cress 16941  +_gcplusg 16962  0_gc0g 17150  Grpcgrp 18577  SubGrpcsubg 18749   GrpAct cga 18895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-subg 18752  df-ga 18896

This theorem is referenced by:  sylow3lem5  19236