Theorem qustgphaus 24088

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 2736	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	qus0 19164	. . . . . . 7 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
4	3	3ad2ant2 1135	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
5		tgpgrp 24043	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
6	5	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
8	7, 2	grpidcl 18941	. . . . . . . 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 7400	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑌) ∈ V
11	10	ecelqsi 8716	. . . . . . 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 2837	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
14	13	snssd 4730	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
16	15	mptpreima 6202	. . . . . 6 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}}
17		nsgsubg 19133	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18	17	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
20	7, 19, 2	eqgid 19155	. . . . . . . . . 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 19103	. . . . . . . . . 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 3956	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25		sseqin2 4163	. . . . . . . 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 19153	. . . . . . . . . . . . 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 8698	. . . . . . . . . . 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 2738	. . . . . . . . . 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 3433	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35		fvex 6853	. . . . . . . . . 10 ⊢ (0g‘𝐺) ∈ V
36	34, 35	elec 8690	. . . . . . . . 9 ⊢ (𝑥 ∈ [(0g‘𝐺)](𝐺 ~QG 𝑌) ↔ (0g‘𝐺)(𝐺 ~QG 𝑌)𝑥)
37		fvex 6853	. . . . . . . . . . 11 ⊢ (0g‘𝐻) ∈ V
38	37	elsn2 4609	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻))
39		eqcom 2743	. . . . . . . . . 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 4166	. . . . . . 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 2779	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}} = 𝑌)
44	16, 43	eqtrid 2783	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = 𝑌)
45		simp3 1139	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
46	44, 45	eqeltrd 2836	. . . 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 24047	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49	48	3ad2ant1 1134	. . . . 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 2737	. . . . . 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 1137	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
54	50, 51, 15, 52, 53	quslem 17507	. . . . 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 23678	. . . . 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 585	. . . 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 714	. . 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 17506	. . . . 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 23685	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
61	60	fveq2d 6844	. . 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 2839	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ∈ (Clsd‘𝐾))
63	1	qustgp 24087	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64	63	3adant3 1133	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65		eqid 2736	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
66	65, 59	tgphaus 24082	. . 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 1087   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  {csn 4567   class class class wbr 5085   ↦ cmpt 5166  ^◡ccnv 5630   “ cima 5634  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367   Er wer 8640  [cec 8641   / cqs 8642  Basecbs 17179  TopOpenctopn 17384  0_gc0g 17402   qTop cqtop 17467   /_s cqus 17469  Grpcgrp 18909  SubGrpcsubg 19096  NrmSGrpcnsg 19097   ~_QG cqg 19098  TopOnctopon 22875  Clsdccld 22981  Hauscha 23273  TopGrpctgp 24036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-ec 8645  df-qs 8649  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-rest 17385  df-topn 17386  df-0g 17404  df-topgen 17406  df-qtop 17471  df-imas 17472  df-qus 17473  df-plusf 18607  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-nsg 19100  df-eqg 19101  df-oppg 19321  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-cn 23192  df-cnp 23193  df-t1 23279  df-haus 23280  df-tx 23527  df-hmeo 23720  df-tmd 24037  df-tgp 24038

This theorem is referenced by: (None)