Theorem subgga 19266

Description: A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
subgga.1	⊢ 𝑋 = (Base‘𝐺)
subgga.2	⊢ + = (+g‘𝐺)
subgga.3	⊢ 𝐻 = (𝐺 ↾s 𝑌)
subgga.4	⊢ 𝐹 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))

Assertion

Ref	Expression
subgga	⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥, + ,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem subgga

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgga.3	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑌)
2	1	subggrp 19096	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3		subgga.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
4	3	fvexi 6848	. . 3 ⊢ 𝑋 ∈ V
5	2, 4	jctir 520	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6		subgrcl 19098	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
8	3	subgss 19094	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
9	8	sselda 3922	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
10	9	adantrr 718	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
11		simprr 773	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
12		subgga.2	. . . . . . . 8 ⊢ + = (+g‘𝐺)
13	3, 12	grpcl 18908	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
14	7, 10, 11, 13	syl3anc 1374	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
15	14	ralrimivva 3181	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 (𝑥 + 𝑦) ∈ 𝑋)
16		subgga.4	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))
17	16	fmpo 8014	. . . . 5 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 (𝑥 + 𝑦) ∈ 𝑋 ↔ 𝐹:(𝑌 × 𝑋)⟶𝑋)
18	15, 17	sylib 218	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
19	1	subgbas 19097	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
20	19	xpeq1d 5653	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
21	20	feq2d 6646	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋 ↔ 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
22	18, 21	mpbid 232	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23		eqid 2737	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
24	23	subg0cl 19101	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
25		oveq12 7369	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑢))
26		ovex 7393	. . . . . . . 8 ⊢ ((0g‘𝐺) + 𝑢) ∈ V
27	25, 16, 26	ovmpoa 7515	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
28	24, 27	sylan 581	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
29	1, 23	subg0 19099	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
30	29	oveq1d 7375	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐻)𝐹𝑢))
31	30	adantr 480	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐻)𝐹𝑢))
32	3, 12, 23	grplid 18934	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
33	6, 32	sylan 581	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
34	28, 31, 33	3eqtr3d 2780	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐻)𝐹𝑢) = 𝑢)
35	6	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝐺 ∈ Grp)
36	8	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
37		simprl 771	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑌)
38	36, 37	sseldd 3923	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
39		simprr 773	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
40	36, 39	sseldd 3923	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑋)
41		simplr 769	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑢 ∈ 𝑋)
42	3, 12	grpass 18909	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
43	35, 38, 40, 41, 42	syl13anc 1375	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
44	3, 12	grpcl 18908	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
45	35, 40, 41, 44	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46		oveq12 7369	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47		ovex 7393	. . . . . . . . . . 11 ⊢ (𝑣 + (𝑤 + 𝑢)) ∈ V
48	46, 16, 47	ovmpoa 7515	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
49	37, 45, 48	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
50	43, 49	eqtr4d 2775	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
51	12	subgcl 19103	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52	51	3expb 1121	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
53	52	adantlr 716	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54		oveq12 7369	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55		ovex 7393	. . . . . . . . . 10 ⊢ ((𝑣 + 𝑤) + 𝑢) ∈ V
56	54, 16, 55	ovmpoa 7515	. . . . . . . . 9 ⊢ (((𝑣 + 𝑤) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
57	53, 41, 56	syl2anc 585	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58		oveq12 7369	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59		ovex 7393	. . . . . . . . . . 11 ⊢ (𝑤 + 𝑢) ∈ V
60	58, 16, 59	ovmpoa 7515	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
61	39, 41, 60	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
62	61	oveq2d 7376	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
63	50, 57, 62	3eqtr4d 2782	. . . . . . 7 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
64	63	ralrimivva 3181	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
65	1, 12	ressplusg 17245	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
66	65	oveqd 7377	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g‘𝐻)𝑤))
67	66	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g‘𝐻)𝑤)𝐹𝑢))
68	67	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
69	19, 68	raleqbidv 3312	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
70	19, 69	raleqbidv 3312	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
71	70	biimpa 476	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ ∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
72	64, 71	syldan 592	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
73	34, 72	jca 511	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → (((0g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
74	73	ralrimiva 3130	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑢 ∈ 𝑋 (((0g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
75	22, 74	jca 511	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢 ∈ 𝑋 (((0g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))))
76		eqid 2737	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
77		eqid 2737	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
78		eqid 2737	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
79	76, 77, 78	isga 19257	. 2 ⊢ (𝐹 ∈ (𝐻 GrpAct 𝑋) ↔ ((𝐻 ∈ Grp ∧ 𝑋 ∈ V) ∧ (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢 ∈ 𝑋 (((0g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))))
80	5, 75, 79	sylanbrc 584	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17170   ↾_s cress 17191  +_gcplusg 17211  0_gc0g 17393  Grpcgrp 18900  SubGrpcsubg 19087   GrpAct cga 19255

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-subg 19090  df-ga 19256

This theorem is referenced by:  gaid2  19269