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| Mirrors > Home > MPE Home > Th. List > istdrg2 | Structured version Visualization version GIF version | ||
| Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| istdrg2.m | ⊢ 𝑀 = (mulGrp‘𝑅) | 
| istdrg2.b | ⊢ 𝐵 = (Base‘𝑅) | 
| istdrg2.z | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| istdrg2 | ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | istdrg2.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 3 | 1, 2 | istdrg 24174 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | 
| 4 | istdrg2.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | istdrg2.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
| 6 | 4, 2, 5 | isdrng 20733 | . . . . . . . 8 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) | 
| 7 | 6 | simprbi 496 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 })) | 
| 8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 })) | 
| 9 | 8 | oveq2d 7447 | . . . . 5 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀 ↾s (Unit‘𝑅)) = (𝑀 ↾s (𝐵 ∖ { 0 }))) | 
| 10 | 9 | eleq1d 2826 | . . . 4 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | 
| 11 | 10 | pm5.32i 574 | . . 3 ⊢ (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | 
| 12 | df-3an 1089 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | |
| 13 | df-3an 1089 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | |
| 14 | 11, 12, 13 | 3bitr4i 303 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | 
| 15 | 3, 14 | bitri 275 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 0gc0g 17484 mulGrpcmgp 20137 Ringcrg 20230 Unitcui 20355 DivRingcdr 20729 TopGrpctgp 24079 TopRingctrg 24164 TopDRingctdrg 24165 | 
| 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-iota 6514 df-fv 6569 df-ov 7434 df-drng 20731 df-tdrg 24169 | 
| This theorem is referenced by: (None) | 
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