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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 | β’ π = (mulGrpβπ ) |
istdrg.1 | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
tdrgunit | β’ (π β TopDRing β (π βΎs π) β TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istrg.1 | . . 3 β’ π = (mulGrpβπ ) | |
2 | istdrg.1 | . . 3 β’ π = (Unitβπ ) | |
3 | 1, 2 | istdrg 24025 | . 2 β’ (π β TopDRing β (π β TopRing β§ π β DivRing β§ (π βΎs π) β TopGrp)) |
4 | 3 | simp3bi 1144 | 1 β’ (π β TopDRing β (π βΎs π) β TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 βΎs cress 17182 mulGrpcmgp 20039 Unitcui 20257 DivRingcdr 20587 TopGrpctgp 23930 TopRingctrg 24015 TopDRingctdrg 24016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 df-tdrg 24020 |
This theorem is referenced by: invrcn2 24039 |
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