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Theorem subgga 19414

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 19239	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3		subgga.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
4	3	fvexi 6899	. . 3 ⊢ 𝑋 ∈ V
5	2, 4	jctir 530	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6		subgrcl 19241	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	6	adantr 486	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
8	3	subgss 19237	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
9	8	sselda 3938	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
10	9	adantrr 730	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
11		simprr 785	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
12		subgga.2	. . . . . . . 8 ⊢ + = (+_g‘𝐺)
13	3, 12	grpcl 19052	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
14	7, 10, 11, 13	syl3anc 1398	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
15	14	ralrimivva 3210	. . . . 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 8071	. . . . 5 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 (𝑥 + 𝑦) ∈ 𝑋 ↔ 𝐹:(𝑌 × 𝑋)⟶𝑋)
18	15, 17	sylib 221	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
19	1	subgbas 19240	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
20	19	xpeq1d 5692	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
21	20	feq2d 6693	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋 ↔ 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
22	18, 21	mpbid 235	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23		eqid 2765	. . . . . . . 8 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
24	23	subg0cl 19244	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑌)
25		oveq12 7428	. . . . . . . 8 ⊢ ((𝑥 = (0_g‘𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0_g‘𝐺) + 𝑢))
26		ovex 7452	. . . . . . . 8 ⊢ ((0_g‘𝐺) + 𝑢) ∈ V
27	25, 16, 26	ovmpoa 7574	. . . . . . 7 ⊢ (((0_g‘𝐺) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐺)𝐹𝑢) = ((0_g‘𝐺) + 𝑢))
28	24, 27	sylan 592	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐺)𝐹𝑢) = ((0_g‘𝐺) + 𝑢))
29	1, 23	subg0 19242	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) = (0_g‘𝐻))
30	29	oveq1d 7434	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((0_g‘𝐺)𝐹𝑢) = ((0_g‘𝐻)𝐹𝑢))
31	30	adantr 486	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐺)𝐹𝑢) = ((0_g‘𝐻)𝐹𝑢))
32	3, 12, 23	grplid 19078	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐺) + 𝑢) = 𝑢)
33	6, 32	sylan 592	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐺) + 𝑢) = 𝑢)
34	28, 31, 33	3eqtr3d 2808	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ((0_g‘𝐻)𝐹𝑢) = 𝑢)
35	6	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝐺 ∈ Grp)
36	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
37		simprl 783	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑌)
38	36, 37	sseldd 3939	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑣 ∈ 𝑋)
39		simprr 785	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑌)
40	36, 39	sseldd 3939	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑤 ∈ 𝑋)
41		simplr 781	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → 𝑢 ∈ 𝑋)
42	3, 12	grpass 19053	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
43	35, 38, 40, 41, 42	syl13anc 1399	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
44	3, 12	grpcl 19052	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
45	35, 40, 41, 44	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47		ovex 7452	. . . . . . . . . . 11 ⊢ (𝑣 + (𝑤 + 𝑢)) ∈ V
48	46, 16, 47	ovmpoa 7574	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
49	37, 45, 48	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
50	43, 49	eqtr4d 2803	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
51	12	subgcl 19246	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52	51	3expb 1138	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
53	52	adantlr 728	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54		oveq12 7428	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55		ovex 7452	. . . . . . . . . 10 ⊢ ((𝑣 + 𝑤) + 𝑢) ∈ V
56	54, 16, 55	ovmpoa 7574	. . . . . . . . 9 ⊢ (((𝑣 + 𝑤) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
57	53, 41, 56	syl2anc 596	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59		ovex 7452	. . . . . . . . . . 11 ⊢ (𝑤 + 𝑢) ∈ V
60	58, 16, 59	ovmpoa 7574	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
61	39, 41, 60	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
62	61	oveq2d 7435	. . . . . . . 8 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
63	50, 57, 62	3eqtr4d 2810	. . . . . . 7 ⊢ (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) ∧ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
64	63	ralrimivva 3210	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ 𝑌 ∀𝑤 ∈ 𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
65	1, 12	ressplusg 17366	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → + = (+_g‘𝐻))
66	65	oveqd 7436	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+_g‘𝐻)𝑤))
67	66	oveq1d 7434	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+_g‘𝐻)𝑤)𝐹𝑢))
68	67	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
69	19, 68	raleqbidv 3340	. . . . . . . 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 3340	. . . . . . 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 482	. . . . . 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 603	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
73	34, 72	jca 521	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢 ∈ 𝑋) → (((0_g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
74	73	ralrimiva 3159	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑢 ∈ 𝑋 (((0_g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
75	22, 74	jca 521	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢 ∈ 𝑋 (((0_g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))))
76		eqid 2765	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
77		eqid 2765	. . 3 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
78		eqid 2765	. . 3 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
79	76, 77, 78	isga 19405	. 2 ⊢ (𝐹 ∈ (𝐻 GrpAct 𝑋) ↔ ((𝐻 ∈ Grp ∧ 𝑋 ∈ V) ∧ (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢 ∈ 𝑋 (((0_g‘𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+_g‘𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))))
80	5, 75, 79	sylanbrc 595	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 Vcvv 3457 ⊆ wss 3906 × cxp 5661 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 Basecbs 17291 ↾_s cress 17312 +_gcplusg 17332 0_gc0g 17514 Grpcgrp 19044 SubGrpcsubg 19230 GrpAct cga 19403

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-0g 17516 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-grp 19047 df-subg 19233 df-ga 19404

This theorem is used by: gaid2 19417

W3C validator