Theorem qustgphaus 24349

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 2760	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
3	1, 2	qus0 19317	. . . . . . 7 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
4	3	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
5		tgpgrp 24304	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
6	5	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
8	7, 2	grpidcl 19089	. . . . . . . 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 7446	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑌) ∈ V
11	10	ecelqsi 8769	. . . . . . 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 2861	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g‘𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
14	13	snssd 4747	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
16	15	mptpreima 6234	. . . . . 6 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}}
17		nsgsubg 19281	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18	17	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19		eqid 2760	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
20	7, 19, 2	eqgid 19305	. . . . . . . . . 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 19250	. . . . . . . . . 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 3965	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25		sseqin2 4169	. . . . . . . 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 19303	. . . . . . . . . . . . 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 8751	. . . . . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g‘𝐺)](𝐺 ~QG 𝑌) = (0g‘𝐻))
32	31	eqeq1d 2762	. . . . . . . . . 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 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35		fvex 6891	. . . . . . . . . 10 ⊢ (0g‘𝐺) ∈ V
36	34, 35	elec 8743	. . . . . . . . 9 ⊢ (𝑥 ∈ [(0g‘𝐺)](𝐺 ~QG 𝑌) ↔ (0g‘𝐺)(𝐺 ~QG 𝑌)𝑥)
37		fvex 6891	. . . . . . . . . . 11 ⊢ (0g‘𝐻) ∈ V
38	37	elsn2 4626	. . . . . . . . . 10 ⊢ ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g‘𝐻))
39		eqcom 2767	. . . . . . . . . 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 4171	. . . . . . 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 2803	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g‘𝐻)}} = 𝑌)
44	16, 43	eqtrid 2807	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (◡(𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g‘𝐻)}) = 𝑌)
45		simp3 1156	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
46	44, 45	eqeltrd 2860	. . . 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 24308	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49	48	3ad2ant1 1151	. . . . 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 2761	. . . . . 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 1154	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
54	50, 51, 15, 52, 53	quslem 17629	. . . . 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 23939	. . . . 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 596	. . . 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 726	. . 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 17628	. . . . 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 23946	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
61	60	fveq2d 6882	. . 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 2863	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g‘𝐻)} ∈ (Clsd‘𝐾))
63	1	qustgp 24348	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64	63	3adant3 1150	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65		eqid 2760	. . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻)
66	65, 59	tgphaus 24343	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5654   “ cima 5658  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7413   Er wer 8693  [cec 8694   / cqs 8695  Basecbs 17301  TopOpenctopn 17506  0_gc0g 17524   qTop cqtop 17589   /_s cqus 17591  Grpcgrp 19057  SubGrpcsubg 19243  NrmSGrpcnsg 19244   ~_QG cqg 19245  TopOnctopon 23135  Clsdccld 23241  Hauscha 23533  TopGrpctgp 24297

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  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 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-ec 8698  df-qs 8702  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-rest 17507  df-topn 17508  df-0g 17526  df-topgen 17528  df-qtop 17593  df-imas 17594  df-qus 17595  df-plusf 18729  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-grp 19060  df-minusg 19061  df-sbg 19062  df-subg 19246  df-nsg 19247  df-eqg 19248  df-oppg 19473  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-cld 23244  df-cn 23452  df-cnp 23453  df-t1 23539  df-haus 23540  df-tx 23788  df-hmeo 23981  df-tmd 24298  df-tgp 24299

This theorem is used by: (None)