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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 2798 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 22771 | . 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 19499 | . . . . . . . 8 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) |
7 | 6 | simprbi 500 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
8 | 7 | adantl 485 | . . . . . 6 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
9 | 8 | oveq2d 7151 | . . . . 5 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀 ↾s (Unit‘𝑅)) = (𝑀 ↾s (𝐵 ∖ { 0 }))) |
10 | 9 | eleq1d 2874 | . . . 4 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
11 | 10 | pm5.32i 578 | . . 3 ⊢ (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
12 | df-3an 1086 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | |
13 | df-3an 1086 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | |
14 | 11, 12, 13 | 3bitr4i 306 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
15 | 3, 14 | bitri 278 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 0gc0g 16705 mulGrpcmgp 19232 Ringcrg 19290 Unitcui 19385 DivRingcdr 19495 TopGrpctgp 22676 TopRingctrg 22761 TopDRingctdrg 22762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-drng 19497 df-tdrg 22766 |
This theorem is referenced by: (None) |
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