Theorem qustgphaus 24039

Description: The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
qustgp.h	⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
qustgphaus.j	⊢ 𝐽 = (TopOpen‘𝐺)
qustgphaus.k	⊢ 𝐾 = (TopOpen‘𝐻)

Assertion

Ref	Expression
qustgphaus	⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)

Proof of Theorem qustgphaus

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qustgp.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
2		eqid 2731	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	qus0 19102	. . . . . . 7 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
4	3	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
5		tgpgrp 23994	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
6	5	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
8	7, 2	grpidcl 18878	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
9	6, 8	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
10		ovex 7379	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑌) ∈ V
11	10	ecelqsi 8694	. . . . . . 7 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
12	9, 11	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
13	4, 12	eqeltrrd 2832	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
14	13	snssd 4761	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
16	15	mptpreima 6185	. . . . . 6 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}}
17		nsgsubg 19071	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18	17	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19		eqid 2731	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
20	7, 19, 2	eqgid 19093	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = 𝑌)
21	18, 20	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = 𝑌)
22	7	subgss 19040	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
23	18, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
24	21, 23	eqsstrd 3969	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25		sseqin2 4173	. . . . . . . 8 ⊢ ([(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g‘𝐺)](𝐺 ~QG 𝑌)) = [(0g‘𝐺)](𝐺 ~QG 𝑌))
26	24, 25	sylib 218	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g‘𝐺)](𝐺 ~QG 𝑌)) = [(0g‘𝐺)](𝐺 ~QG 𝑌))
27	7, 19	eqger 19091	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
28	18, 27	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
29	28, 9	erth 8676	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((0g‘𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g‘𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
30	29	adantr 480	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g‘𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g‘𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
31	4	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
32	31	eqeq1d 2733	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ([(0g‘𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌) ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌)))
33	30, 32	bitrd 279	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g‘𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌)))
34		vex 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35		fvex 6835	. . . . . . . . . 10 ⊢ (0g‘𝐺) ∈ V
36	34, 35	elec 8668	. . . . . . . . 9 ⊢ (𝑥 ∈ [(0g‘𝐺)](𝐺 ~QG 𝑌) ↔ (0g‘𝐺)(𝐺 ~QG 𝑌)𝑥)
37		fvex 6835	. . . . . . . . . . 11 ⊢ (0g‘𝐻) ∈ V
38	37	elsn2 4618	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻))
39		eqcom 2738	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻) ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌))
40	38, 39	bitri 275	. . . . . . . . 9 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌))
41	33, 36, 40	3bitr4g 314	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥 ∈ [(0g‘𝐺)](𝐺 ~QG 𝑌) ↔ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}))
42	41	rabbi2dva 4176	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g‘𝐺)](𝐺 ~QG 𝑌)) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}})
43	26, 42, 21	3eqtr3d 2774	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}} = 𝑌)
44	16, 43	eqtrid 2778	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = 𝑌)
45		simp3 1138	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
46	44, 45	eqeltrd 2831	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) ∈ (Clsd‘𝐽))
47		qustgphaus.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐺)
48	47, 7	tgptopon 23998	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49	48	3ad2ant1 1133	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
50	1	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)))
51		eqidd 2732	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Base‘𝐺) = (Base‘𝐺))
52	10	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐺 ~QG 𝑌) ∈ V)
53		simp1 1136	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
54	50, 51, 15, 52, 53	quslem 17447	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)):(Base‘𝐺)–onto→((Base‘𝐺) / (𝐺 ~QG 𝑌)))
55		qtopcld 23629	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)):(Base‘𝐺)–onto→((Base‘𝐺) / (𝐺 ~QG 𝑌))) → ({(0g‘𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))) ↔ ({(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)) ∧ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) ∈ (Clsd‘𝐽))))
56	49, 54, 55	syl2anc 584	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ({(0g‘𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))) ↔ ({(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)) ∧ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) ∈ (Clsd‘𝐽))))
57	14, 46, 56	mpbir2and 713	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
58	50, 51, 15, 52, 53	qusval 17446	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 = ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “s 𝐺))
59		qustgphaus.k	. . . . 5 ⊢ 𝐾 = (TopOpen‘𝐻)
60	58, 51, 54, 53, 47, 59	imastopn 23636	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
61	60	fveq2d 6826	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
62	57, 61	eleqtrrd 2834	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ∈ (Clsd‘𝐾))
63	1	qustgp 24038	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64	63	3adant3 1132	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65		eqid 2731	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
66	65, 59	tgphaus 24033	. . 3 ⊢ (𝐻 ∈ TopGrp → (𝐾 ∈ Haus ↔ {(0g‘𝐻)} ∈ (Clsd‘𝐾)))
67	64, 66	syl 17	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐾 ∈ Haus ↔ {(0g‘𝐻)} ∈ (Clsd‘𝐾)))
68	62, 67	mpbird 257	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  {csn 4576   class class class wbr 5091   ↦ cmpt 5172  ^◡ccnv 5615   “ cima 5619  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346   Er wer 8619  [cec 8620   / cqs 8621  Basecbs 17120  TopOpenctopn 17325  0_gc0g 17343   qTop cqtop 17407   /_s cqus 17409  Grpcgrp 18846  SubGrpcsubg 19033  NrmSGrpcnsg 19034   ~_QG cqg 19035  TopOnctopon 22826  Clsdccld 22932  Hauscha 23224  TopGrpctgp 23987

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-ec 8624  df-qs 8628  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-rest 17326  df-topn 17327  df-0g 17345  df-topgen 17347  df-qtop 17411  df-imas 17412  df-qus 17413  df-plusf 18547  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-minusg 18850  df-sbg 18851  df-subg 19036  df-nsg 19037  df-eqg 19038  df-oppg 19259  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cld 22935  df-cn 23143  df-cnp 23144  df-t1 23230  df-haus 23231  df-tx 23478  df-hmeo 23671  df-tmd 23988  df-tgp 23989

This theorem is referenced by: (None)