![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isngp | Structured version Visualization version GIF version |
Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
isngp.n | ⊢ 𝑁 = (norm‘𝐺) |
isngp.z | ⊢ − = (-g‘𝐺) |
isngp.d | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
isngp | ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3979 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp)) | |
2 | 1 | anbi1i 624 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺)) | |
4 | isngp.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁) |
6 | fveq2 6907 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺)) | |
7 | isngp.z | . . . . . 6 ⊢ − = (-g‘𝐺) | |
8 | 6, 7 | eqtr4di 2793 | . . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − ) |
9 | 5, 8 | coeq12d 5878 | . . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − )) |
10 | fveq2 6907 | . . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺)) | |
11 | isngp.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐺) | |
12 | 10, 11 | eqtr4di 2793 | . . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷) |
13 | 9, 12 | sseq12d 4029 | . . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷)) |
14 | df-ngp 24612 | . . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} | |
15 | 13, 14 | elrab2 3698 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
16 | df-3an 1088 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) | |
17 | 2, 15, 16 | 3bitr4i 303 | 1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 ∘ ccom 5693 ‘cfv 6563 distcds 17307 Grpcgrp 18964 -gcsg 18966 MetSpcms 24344 normcnm 24605 NrmGrpcngp 24606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-co 5698 df-iota 6516 df-fv 6571 df-ngp 24612 |
This theorem is referenced by: isngp2 24626 ngpgrp 24628 ngpms 24629 tngngp2 24689 cnngp 24816 zhmnrg 33928 |
Copyright terms: Public domain | W3C validator |