Theorem subgga 19251

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 19084	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3		subgga.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
4	3	fvexi 6911	. . 3 ⊢ 𝑋 ∈ V
5	2, 4	jctir 520	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6		subgrcl 19086	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
8	3	subgss 19082	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
9	8	sselda 3980	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
10	9	adantrr 716	. . . . . . 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 18898	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
14	7, 10, 11, 13	syl3anc 1369	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
15	14	ralrimivva 3197	. . . . 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 8072	. . . . 5 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 (𝑥 + 𝑦) ∈ 𝑋 ↔ 𝐹:(𝑌 × 𝑋)⟶𝑋)
18	15, 17	sylib 217	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
19	1	subgbas 19085	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
20	19	xpeq1d 5707	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
21	20	feq2d 6708	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋 ↔ 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
22	18, 21	mpbid 231	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23		eqid 2728	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
24	23	subg0cl 19089	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
25		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑢))
26		ovex 7453	. . . . . . . 8 ⊢ ((0g‘𝐺) + 𝑢) ∈ V
27	25, 16, 26	ovmpoa 7576	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
28	24, 27	sylan 579	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺)𝐹𝑢) = ((0g‘𝐺) + 𝑢))
29	1, 23	subg0 19087	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
30	29	oveq1d 7435	. . . . . . 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 18924	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
33	6, 32	sylan 579	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + 𝑢) = 𝑢)
34	28, 31, 33	3eqtr3d 2776	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐻)𝐹𝑢) = 𝑢)
35	6	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝐺 ∈ Grp)
36	8	ad2antrr 725	. . . . . . . . . . 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 3981	. . . . . . . . . 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 3981	. . . . . . . . . 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 18899	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
43	35, 38, 40, 41, 42	syl13anc 1370	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
44	3, 12	grpcl 18898	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
45	35, 40, 41, 44	syl3anc 1369	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46		oveq12 7429	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑣 + (𝑤 + 𝑢)) ∈ V
48	46, 16, 47	ovmpoa 7576	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
49	37, 45, 48	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
50	43, 49	eqtr4d 2771	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
51	12	subgcl 19091	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52	51	3expb 1118	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
53	52	adantlr 714	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54		oveq12 7429	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55		ovex 7453	. . . . . . . . . 10 ⊢ ((𝑣 + 𝑤) + 𝑢) ∈ V
56	54, 16, 55	ovmpoa 7576	. . . . . . . . 9 ⊢ (((𝑣 + 𝑤) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
57	53, 41, 56	syl2anc 583	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58		oveq12 7429	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑤 + 𝑢) ∈ V
60	58, 16, 59	ovmpoa 7576	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
61	39, 41, 60	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
62	61	oveq2d 7436	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
63	50, 57, 62	3eqtr4d 2778	. . . . . . 7 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
64	63	ralrimivva 3197	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
65	1, 12	ressplusg 17271	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
66	65	oveqd 7437	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g‘𝐻)𝑤))
67	66	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g‘𝐻)𝑤)𝐹𝑢))
68	67	eqeq1d 2730	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
69	19, 68	raleqbidv 3339	. . . . . . . 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 3339	. . . . . . 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 590	. . . . 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 3143	. . 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 2728	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
77		eqid 2728	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
78		eqid 2728	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
79	76, 77, 78	isga 19242	. 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 582	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471   ⊆ wss 3947   × cxp 5676  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17180   ↾_s cress 17209  +_gcplusg 17233  0_gc0g 17421  Grpcgrp 18890  SubGrpcsubg 19075   GrpAct cga 19240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-subg 19078  df-ga 19241

This theorem is referenced by:  gaid2  19254