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| 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 3967 | . . 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 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺)) | |
| 4 | isngp.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁) | 
| 6 | fveq2 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺)) | |
| 7 | isngp.z | . . . . . 6 ⊢ − = (-g‘𝐺) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − ) | 
| 9 | 5, 8 | coeq12d 5875 | . . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − )) | 
| 10 | fveq2 6906 | . . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺)) | |
| 11 | isngp.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐺) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷) | 
| 13 | 9, 12 | sseq12d 4017 | . . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷)) | 
| 14 | df-ngp 24596 | . . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} | |
| 15 | 13, 14 | elrab2 3695 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷)) | 
| 16 | df-3an 1089 | . 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 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∘ ccom 5689 ‘cfv 6561 distcds 17306 Grpcgrp 18951 -gcsg 18953 MetSpcms 24328 normcnm 24589 NrmGrpcngp 24590 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-co 5694 df-iota 6514 df-fv 6569 df-ngp 24596 | 
| This theorem is referenced by: isngp2 24610 ngpgrp 24612 ngpms 24613 tngngp2 24673 cnngp 24800 zhmnrg 33966 | 
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