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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 4166 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp)) | |
2 | 1 | anbi1i 623 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
3 | fveq2 6663 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺)) | |
4 | isngp.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
5 | 3, 4 | syl6eqr 2871 | . . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁) |
6 | fveq2 6663 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺)) | |
7 | isngp.z | . . . . . 6 ⊢ − = (-g‘𝐺) | |
8 | 6, 7 | syl6eqr 2871 | . . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − ) |
9 | 5, 8 | coeq12d 5728 | . . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − )) |
10 | fveq2 6663 | . . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺)) | |
11 | isngp.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐺) | |
12 | 10, 11 | syl6eqr 2871 | . . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷) |
13 | 9, 12 | sseq12d 3997 | . . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷)) |
14 | df-ngp 23120 | . . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} | |
15 | 13, 14 | elrab2 3680 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
16 | df-3an 1081 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) | |
17 | 2, 15, 16 | 3bitr4i 304 | 1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∘ ccom 5552 ‘cfv 6348 distcds 16562 Grpcgrp 18041 -gcsg 18043 MetSpcms 22855 normcnm 23113 NrmGrpcngp 23114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-co 5557 df-iota 6307 df-fv 6356 df-ngp 23120 |
This theorem is referenced by: isngp2 23133 ngpgrp 23135 ngpms 23136 tngngp2 23188 cnngp 23315 zhmnrg 31107 |
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