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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 2735 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 24190 | . 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 20750 | . . . . . . . 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 2824 | . . . 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 1088 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | |
13 | df-3an 1088 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 0gc0g 17486 mulGrpcmgp 20152 Ringcrg 20251 Unitcui 20372 DivRingcdr 20746 TopGrpctgp 24095 TopRingctrg 24180 TopDRingctdrg 24181 |
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-iota 6516 df-fv 6571 df-ov 7434 df-drng 20748 df-tdrg 24185 |
This theorem is referenced by: (None) |
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