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Mirrors > Home > MPE Home > Th. List > tdrgunit | Structured version Visualization version GIF version |
Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istrg.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
istdrg.1 | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
tdrgunit | ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istrg.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | istdrg.1 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | istdrg 22768 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) |
4 | 3 | simp3bi 1143 | 1 ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ↾s cress 16478 mulGrpcmgp 19233 Unitcui 19383 DivRingcdr 19496 TopGrpctgp 22673 TopRingctrg 22758 TopDRingctdrg 22759 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-tdrg 22763 |
This theorem is referenced by: invrcn2 22782 |
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