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 6841	. . 3 ⊢ 𝑋 ∈ V
5	2, 4	jctir 525	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6		subgrcl 19098	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	6	adantr 481	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
8	3	subgss 19094	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
9	8	sselda 3915	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
10	9	adantrr 723	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
11		simprr 778	. . . . . . 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 1379	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
15	14	ralrimivva 3182	. . . . 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 8010	. . . . 5 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 (𝑥 + 𝑦) ∈ 𝑋 ↔ 𝐹:(𝑌 × 𝑋)⟶𝑋)
18	15, 17	sylib 219	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
19	1	subgbas 19097	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
20	19	xpeq1d 5647	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
21	20	feq2d 6639	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋 ↔ 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
22	18, 21	mpbid 233	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23		eqid 2739	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
24	23	subg0cl 19101	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
25		oveq12 7365	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑢))
26		ovex 7389	. . . . . . . 8 ⊢ ((0g‘𝐺) + 𝑢) ∈ V
27	25, 16, 26	ovmpoa 7511	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
28	24, 27	sylan 586	. . . . . 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 7371	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐻)𝐹𝑢))
31	30	adantr 481	. . . . . 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 586	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
34	28, 31, 33	3eqtr3d 2782	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐻)𝐹𝑢) = 𝑢)
35	6	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝐺 ∈ Grp)
36	8	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
37		simprl 776	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑌)
38	36, 37	sseldd 3916	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
39		simprr 778	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
40	36, 39	sseldd 3916	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑋)
41		simplr 774	. . . . . . . . . 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 1380	. . . . . . . . 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 1379	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46		oveq12 7365	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47		ovex 7389	. . . . . . . . . . 11 ⊢ (𝑣 + (𝑤 + 𝑢)) ∈ V
48	46, 16, 47	ovmpoa 7511	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
49	37, 45, 48	syl2anc 590	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
50	43, 49	eqtr4d 2777	. . . . . . . 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 1126	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
53	52	adantlr 721	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54		oveq12 7365	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55		ovex 7389	. . . . . . . . . 10 ⊢ ((𝑣 + 𝑤) + 𝑢) ∈ V
56	54, 16, 55	ovmpoa 7511	. . . . . . . . 9 ⊢ (((𝑣 + 𝑤) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
57	53, 41, 56	syl2anc 590	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58		oveq12 7365	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59		ovex 7389	. . . . . . . . . . 11 ⊢ (𝑤 + 𝑢) ∈ V
60	58, 16, 59	ovmpoa 7511	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
61	39, 41, 60	syl2anc 590	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
62	61	oveq2d 7372	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
63	50, 57, 62	3eqtr4d 2784	. . . . . . 7 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
64	63	ralrimivva 3182	. . . . . 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 7373	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g‘𝐻)𝑤))
67	66	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g‘𝐻)𝑤)𝐹𝑢))
68	67	eqeq1d 2741	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
69	19, 68	raleqbidv 3313	. . . . . . . 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 3313	. . . . . . 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 477	. . . . . 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 597	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
73	34, 72	jca 516	. . . 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 3131	. . 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 516	. 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 2739	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
77		eqid 2739	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
78		eqid 2739	. . 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 589	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883   × cxp 5616  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  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 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  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 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  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