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Mirrors > Home > MPE Home > Th. List > tdrgdrng | Structured version Visualization version GIF version |
Description: A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tdrgdrng | ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2798 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 22771 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp)) |
4 | 3 | simp2bi 1143 | 1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ↾s cress 16476 mulGrpcmgp 19232 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-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-tdrg 22766 |
This theorem is referenced by: tvclvec 22804 |
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