Definition df-tmd 23567

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](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}

Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-tmd

Step	Hyp	Ref	Expression
1		ctmd 23565	. 2 class TopMnd
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1540	. . . . . 6 class 𝑓
4		cplusf 18554	. . . . . 6 class +𝑓
5	3, 4	cfv 6540	. . . . 5 class (+𝑓‘𝑓)
6		vj	. . . . . . . 8 setvar 𝑗
7	6	cv 1540	. . . . . . 7 class 𝑗
8		ctx 23055	. . . . . . 7 class ×t
9	7, 7, 8	co 7405	. . . . . 6 class (𝑗 ×t 𝑗)
10		ccn 22719	. . . . . 6 class Cn
11	9, 7, 10	co 7405	. . . . 5 class ((𝑗 ×t 𝑗) Cn 𝑗)
12	5, 11	wcel 2106	. . . 4 wff (+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)
13		ctopn 17363	. . . . 5 class TopOpen
14	3, 13	cfv 6540	. . . 4 class (TopOpen‘𝑓)
15	12, 6, 14	wsbc 3776	. . 3 wff [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)
16		cmnd 18621	. . . 4 class Mnd
17		ctps 22425	. . . 4 class TopSp
18	16, 17	cin 3946	. . 3 class (Mnd ∩ TopSp)
19	15, 2, 18	crab 3432	. 2 class {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
20	1, 19	wceq 1541	1 wff TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}

This definition is referenced by:  istmd  23569