Definition df-tgp 23568

Description: Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)

Assertion

Ref	Expression
df-tgp	⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}

Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-tgp

Step	Hyp	Ref	Expression
1		ctgp 23566	. 2 class TopGrp
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1540	. . . . . 6 class 𝑓
4		cminusg 18816	. . . . . 6 class invg
5	3, 4	cfv 6540	. . . . 5 class (invg‘𝑓)
6		vj	. . . . . . 7 setvar 𝑗
7	6	cv 1540	. . . . . 6 class 𝑗
8		ccn 22719	. . . . . 6 class Cn
9	7, 7, 8	co 7405	. . . . 5 class (𝑗 Cn 𝑗)
10	5, 9	wcel 2106	. . . 4 wff (invg‘𝑓) ∈ (𝑗 Cn 𝑗)
11		ctopn 17363	. . . . 5 class TopOpen
12	3, 11	cfv 6540	. . . 4 class (TopOpen‘𝑓)
13	10, 6, 12	wsbc 3776	. . 3 wff [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)
14		cgrp 18815	. . . 4 class Grp
15		ctmd 23565	. . . 4 class TopMnd
16	14, 15	cin 3946	. . 3 class (Grp ∩ TopMnd)
17	13, 2, 16	crab 3432	. 2 class {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
18	1, 17	wceq 1541	1 wff TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}

This definition is referenced by:  istgp  23572