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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 3758 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp)) | |
2 | 1 | anbi1i 727 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺)) | |
4 | isngp.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
5 | 3, 4 | syl6eqr 2662 | . . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁) |
6 | fveq2 6103 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺)) | |
7 | isngp.z | . . . . . 6 ⊢ − = (-g‘𝐺) | |
8 | 6, 7 | syl6eqr 2662 | . . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − ) |
9 | 5, 8 | coeq12d 5208 | . . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − )) |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺)) | |
11 | isngp.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐺) | |
12 | 10, 11 | syl6eqr 2662 | . . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷) |
13 | 9, 12 | sseq12d 3597 | . . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷)) |
14 | df-ngp 22198 | . . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} | |
15 | 13, 14 | elrab2 3333 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
16 | df-3an 1033 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) | |
17 | 2, 15, 16 | 3bitr4i 291 | 1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ∘ ccom 5042 ‘cfv 5804 distcds 15777 Grpcgrp 17245 -gcsg 17247 MetSpcmt 21933 normcnm 22191 NrmGrpcngp 22192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-co 5047 df-iota 5768 df-fv 5812 df-ngp 22198 |
This theorem is referenced by: isngp2 22211 ngpgrp 22213 ngpms 22214 tngngp2 22266 cnngp 22393 zhmnrg 29339 |
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