Theorem subgga 19229

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 19059	. . 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 19061	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
8	3	subgss 19057	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
9	8	sselda 3933	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
10	9	adantrr 717	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
11		simprr 772	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
12		subgga.2	. . . . . . . 8 ⊢ + = (+g‘𝐺)
13	3, 12	grpcl 18871	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
14	7, 10, 11, 13	syl3anc 1373	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
15	14	ralrimivva 3179	. . . . 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 8012	. . . . 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 19060	. . . . . 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 2736	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
24	23	subg0cl 19064	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
25		oveq12 7367	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑢))
26		ovex 7391	. . . . . . . 8 ⊢ ((0g‘𝐺) + 𝑢) ∈ V
27	25, 16, 26	ovmpoa 7513	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
28	24, 27	sylan 580	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
29	1, 23	subg0 19062	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
30	29	oveq1d 7373	. . . . . . 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 18897	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
33	6, 32	sylan 580	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
34	28, 31, 33	3eqtr3d 2779	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐻)𝐹𝑢) = 𝑢)
35	6	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝐺 ∈ Grp)
36	8	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
37		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑌)
38	36, 37	sseldd 3934	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
39		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
40	36, 39	sseldd 3934	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑋)
41		simplr 768	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑢 ∈ 𝑋)
42	3, 12	grpass 18872	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
43	35, 38, 40, 41, 42	syl13anc 1374	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
44	3, 12	grpcl 18871	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
45	35, 40, 41, 44	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46		oveq12 7367	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47		ovex 7391	. . . . . . . . . . 11 ⊢ (𝑣 + (𝑤 + 𝑢)) ∈ V
48	46, 16, 47	ovmpoa 7513	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
49	37, 45, 48	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
50	43, 49	eqtr4d 2774	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
51	12	subgcl 19066	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52	51	3expb 1120	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
53	52	adantlr 715	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54		oveq12 7367	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55		ovex 7391	. . . . . . . . . 10 ⊢ ((𝑣 + 𝑤) + 𝑢) ∈ V
56	54, 16, 55	ovmpoa 7513	. . . . . . . . 9 ⊢ (((𝑣 + 𝑤) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
57	53, 41, 56	syl2anc 584	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58		oveq12 7367	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59		ovex 7391	. . . . . . . . . . 11 ⊢ (𝑤 + 𝑢) ∈ V
60	58, 16, 59	ovmpoa 7513	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
61	39, 41, 60	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
62	61	oveq2d 7374	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
63	50, 57, 62	3eqtr4d 2781	. . . . . . 7 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
64	63	ralrimivva 3179	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
65	1, 12	ressplusg 17211	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
66	65	oveqd 7375	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g‘𝐻)𝑤))
67	66	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g‘𝐻)𝑤)𝐹𝑢))
68	67	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
69	19, 68	raleqbidv 3316	. . . . . . . 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 3316	. . . . . . 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 591	. . . . 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 3128	. . 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 2736	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
77		eqid 2736	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
78		eqid 2736	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
79	76, 77, 78	isga 19220	. 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 583	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136   ↾_s cress 17157  +_gcplusg 17177  0_gc0g 17359  Grpcgrp 18863  SubGrpcsubg 19050   GrpAct cga 19218

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-subg 19053  df-ga 19219

This theorem is referenced by:  gaid2  19232