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

Definition df-tmd 21686
 Description: Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
df-tmd TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-tmd
StepHypRef Expression
1 ctmd 21684 . 2 class TopMnd
2 vf . . . . . . 7 setvar 𝑓
32cv 1474 . . . . . 6 class 𝑓
4 cplusf 17062 . . . . . 6 class +𝑓
53, 4cfv 5804 . . . . 5 class (+𝑓𝑓)
6 vj . . . . . . . 8 setvar 𝑗
76cv 1474 . . . . . . 7 class 𝑗
8 ctx 21173 . . . . . . 7 class ×t
97, 7, 8co 6549 . . . . . 6 class (𝑗 ×t 𝑗)
10 ccn 20838 . . . . . 6 class Cn
119, 7, 10co 6549 . . . . 5 class ((𝑗 ×t 𝑗) Cn 𝑗)
125, 11wcel 1977 . . . 4 wff (+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)
13 ctopn 15905 . . . . 5 class TopOpen
143, 13cfv 5804 . . . 4 class (TopOpen‘𝑓)
1512, 6, 14wsbc 3402 . . 3 wff [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)
16 cmnd 17117 . . . 4 class Mnd
17 ctps 20519 . . . 4 class TopSp
1816, 17cin 3539 . . 3 class (Mnd ∩ TopSp)
1915, 2, 18crab 2900 . 2 class {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
201, 19wceq 1475 1 wff TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
 Colors of variables: wff setvar class This definition is referenced by:  istmd  21688
 Copyright terms: Public domain W3C validator