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Definition df-tng 22140
Description: Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
df-tng toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
Distinct variable group:   𝑓,𝑔

Detailed syntax breakdown of Definition df-tng
StepHypRef Expression
1 ctng 22134 . 2 class toNrmGrp
2 vg . . 3 setvar 𝑔
3 vf . . 3 setvar 𝑓
4 cvv 3172 . . 3 class V
52cv 1473 . . . . 5 class 𝑔
6 cnx 15638 . . . . . . 7 class ndx
7 cds 15723 . . . . . . 7 class dist
86, 7cfv 5790 . . . . . 6 class (dist‘ndx)
93cv 1473 . . . . . . 7 class 𝑓
10 csg 17193 . . . . . . . 8 class -g
115, 10cfv 5790 . . . . . . 7 class (-g𝑔)
129, 11ccom 5032 . . . . . 6 class (𝑓 ∘ (-g𝑔))
138, 12cop 4130 . . . . 5 class ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩
14 csts 15639 . . . . 5 class sSet
155, 13, 14co 6527 . . . 4 class (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩)
16 cts 15720 . . . . . 6 class TopSet
176, 16cfv 5790 . . . . 5 class (TopSet‘ndx)
18 cmopn 19503 . . . . . 6 class MetOpen
1912, 18cfv 5790 . . . . 5 class (MetOpen‘(𝑓 ∘ (-g𝑔)))
2017, 19cop 4130 . . . 4 class ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩
2115, 20, 14co 6527 . . 3 class ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩)
222, 3, 4, 4, 21cmpt2 6529 . 2 class (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
231, 22wceq 1474 1 wff toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
Colors of variables: wff setvar class
This definition is referenced by:  reldmtng  22190  tngval  22191
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