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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 2825 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2825 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 22346 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp)) |
4 | 3 | simp1bi 1179 | 1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 ↾s cress 16230 mulGrpcmgp 18850 Unitcui 19000 DivRingcdr 19110 TopGrpctgp 22252 TopRingctrg 22336 TopDRingctdrg 22337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-iota 6090 df-fv 6135 df-ov 6913 df-tdrg 22341 |
This theorem is referenced by: tdrgring 22355 tdrgtmd 22356 tdrgtps 22357 dvrcn 22364 |
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