Theorem qustgphaus 24245

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 2769	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	qus0 19256	. . . . . . 7 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
4	3	3ad2ant2 1150	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
5		tgpgrp 24200	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
6	5	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7		eqid 2769	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
8	7, 2	grpidcl 19028	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
9	6, 8	syl 18	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
10		ovex 7441	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑌) ∈ V
11	10	ecelqsi 8763	. . . . . . 7 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
12	9, 11	syl 18	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
13	4, 12	eqeltrrd 2870	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
14	13	snssd 4754	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15		eqid 2769	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
16	15	mptpreima 6236	. . . . . 6 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}}
17		nsgsubg 19220	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18	17	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19		eqid 2769	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
20	7, 19, 2	eqgid 19244	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = 𝑌)
21	18, 20	syl 18	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = 𝑌)
22	7	subgss 19189	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
23	18, 22	syl 18	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
24	21, 23	eqsstrd 3979	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25		sseqin2 4184	. . . . . . . 8 ⊢ ([(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g‘𝐺)](𝐺 ~QG 𝑌)) = [(0g‘𝐺)](𝐺 ~QG 𝑌))
26	24, 25	sylib 221	. . . . . . 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 19242	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
28	18, 27	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
29	28, 9	erth 8745	. . . . . . . . . . 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 485	. . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
32	31	eqeq1d 2771	. . . . . . . . . 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 282	. . . . . . . . 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 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35		fvex 6892	. . . . . . . . . 10 ⊢ (0g‘𝐺) ∈ V
36	34, 35	elec 8737	. . . . . . . . 9 ⊢ (𝑥 ∈ [(0g‘𝐺)](𝐺 ~QG 𝑌) ↔ (0g‘𝐺)(𝐺 ~QG 𝑌)𝑥)
37		fvex 6892	. . . . . . . . . . 11 ⊢ (0g‘𝐻) ∈ V
38	37	elsn2 4633	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻))
39		eqcom 2776	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻) ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌))
40	38, 39	bitri 278	. . . . . . . . 9 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ (0g‘𝐻) = [𝑥](𝐺 ~QG 𝑌))
41	33, 36, 40	3bitr4g 317	. . . . . . . 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 4186	. . . . . . 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 2812	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}} = 𝑌)
44	16, 43	eqtrid 2816	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = 𝑌)
45		simp3 1154	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
46	44, 45	eqeltrd 2869	. . . 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 24204	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49	48	3ad2ant1 1149	. . . . 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 2770	. . . . . 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 1152	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
54	50, 51, 15, 52, 53	quslem 17593	. . . . 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 23835	. . . . 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 595	. . . 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 725	. . 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 17592	. . . . 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 23842	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
61	60	fveq2d 6883	. . 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 2872	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ∈ (Clsd‘𝐾))
63	1	qustgp 24244	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64	63	3adant3 1148	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65		eqid 2769	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
66	65, 59	tgphaus 24239	. . 3 ⊢ (𝐻 ∈ TopGrp → (𝐾 ∈ Haus ↔ {(0g‘𝐻)} ∈ (Clsd‘𝐾)))
67	64, 66	syl 18	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐾 ∈ Haus ↔ {(0g‘𝐻)} ∈ (Clsd‘𝐾)))
68	62, 67	mpbird 260	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  {csn 4591   class class class wbr 5110   ↦ cmpt 5193  ^◡ccnv 5658   “ cima 5662  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7408   Er wer 8687  [cec 8688   / cqs 8689  Basecbs 17265  TopOpenctopn 17470  0_gc0g 17488   qTop cqtop 17553   /_s cqus 17555  Grpcgrp 18996  SubGrpcsubg 19182  NrmSGrpcnsg 19183   ~_QG cqg 19184  TopOnctopon 23032  Clsdccld 23138  Hauscha 23430  TopGrpctgp 24193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-rest 17471  df-topn 17472  df-0g 17490  df-topgen 17492  df-qtop 17557  df-imas 17558  df-qus 17559  df-plusf 18693  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-grp 18999  df-minusg 19000  df-sbg 19001  df-subg 19185  df-nsg 19186  df-eqg 19187  df-oppg 19412  df-top 23016  df-topon 23033  df-topsp 23055  df-bases 23068  df-cld 23141  df-cn 23349  df-cnp 23350  df-t1 23436  df-haus 23437  df-tx 23684  df-hmeo 23877  df-tmd 24194  df-tgp 24195

This theorem is referenced by: (None)