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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 2821 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 22757 | . 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 19489 | . . . . . . . 8 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) |
7 | 6 | simprbi 499 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
8 | 7 | adantl 484 | . . . . . 6 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
9 | 8 | oveq2d 7158 | . . . . 5 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀 ↾s (Unit‘𝑅)) = (𝑀 ↾s (𝐵 ∖ { 0 }))) |
10 | 9 | eleq1d 2897 | . . . 4 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
11 | 10 | pm5.32i 577 | . . 3 ⊢ (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
12 | df-3an 1085 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp)) | |
13 | df-3an 1085 | . . 3 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | |
14 | 11, 12, 13 | 3bitr4i 305 | . 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
15 | 3, 14 | bitri 277 | 1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3921 {csn 4553 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 ↾s cress 16467 0gc0g 16696 mulGrpcmgp 19222 Ringcrg 19280 Unitcui 19372 DivRingcdr 19485 TopGrpctgp 22662 TopRingctrg 22747 TopDRingctdrg 22748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-iota 6300 df-fv 6349 df-ov 7145 df-drng 19487 df-tdrg 22752 |
This theorem is referenced by: (None) |
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