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Mirrors > Home > MPE Home > Th. List > tdrgtrg | Structured version Visualization version GIF version |
Description: A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tdrgtrg | ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2735 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 24190 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp)) |
4 | 3 | simp1bi 1144 | 1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 mulGrpcmgp 20152 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-tdrg 24185 |
This theorem is referenced by: tdrgring 24199 tdrgtmd 24200 tdrgtps 24201 dvrcn 24208 |
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