Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-tgp Structured version   Visualization version   GIF version

Definition df-tgp 21687
 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](invg𝑓) ∈ (𝑗 Cn 𝑗)}
Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-tgp
StepHypRef Expression
1 ctgp 21685 . 2 class TopGrp
2 vf . . . . . . 7 setvar 𝑓
32cv 1474 . . . . . 6 class 𝑓
4 cminusg 17246 . . . . . 6 class invg
53, 4cfv 5804 . . . . 5 class (invg𝑓)
6 vj . . . . . . 7 setvar 𝑗
76cv 1474 . . . . . 6 class 𝑗
8 ccn 20838 . . . . . 6 class Cn
97, 7, 8co 6549 . . . . 5 class (𝑗 Cn 𝑗)
105, 9wcel 1977 . . . 4 wff (invg𝑓) ∈ (𝑗 Cn 𝑗)
11 ctopn 15905 . . . . 5 class TopOpen
123, 11cfv 5804 . . . 4 class (TopOpen‘𝑓)
1310, 6, 12wsbc 3402 . . 3 wff [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)
14 cgrp 17245 . . . 4 class Grp
15 ctmd 21684 . . . 4 class TopMnd
1614, 15cin 3539 . . 3 class (Grp ∩ TopMnd)
1713, 2, 16crab 2900 . 2 class {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
181, 17wceq 1475 1 wff TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
 Colors of variables: wff setvar class This definition is referenced by:  istgp  21691
 Copyright terms: Public domain W3C validator