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Mirrors > Home > MPE Home > Th. List > df-ngp | Structured version Visualization version GIF version |
Description: Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
df-ngp | ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cngp 23714 | . 2 class NrmGrp | |
2 | vg | . . . . . . 7 setvar 𝑔 | |
3 | 2 | cv 1540 | . . . . . 6 class 𝑔 |
4 | cnm 23713 | . . . . . 6 class norm | |
5 | 3, 4 | cfv 6430 | . . . . 5 class (norm‘𝑔) |
6 | csg 18560 | . . . . . 6 class -g | |
7 | 3, 6 | cfv 6430 | . . . . 5 class (-g‘𝑔) |
8 | 5, 7 | ccom 5592 | . . . 4 class ((norm‘𝑔) ∘ (-g‘𝑔)) |
9 | cds 16952 | . . . . 5 class dist | |
10 | 3, 9 | cfv 6430 | . . . 4 class (dist‘𝑔) |
11 | 8, 10 | wss 3891 | . . 3 wff ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) |
12 | cgrp 18558 | . . . 4 class Grp | |
13 | cms 23452 | . . . 4 class MetSp | |
14 | 12, 13 | cin 3890 | . . 3 class (Grp ∩ MetSp) |
15 | 11, 2, 14 | crab 3069 | . 2 class {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} |
16 | 1, 15 | wceq 1541 | 1 wff NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} |
Colors of variables: wff setvar class |
This definition is referenced by: isngp 23733 |
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