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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 24098 | . 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 6553 (class class class)co 7426 βΎs cress 17218 mulGrpcmgp 20088 Unitcui 20308 DivRingcdr 20638 TopGrpctgp 24003 TopRingctrg 24088 TopDRingctdrg 24089 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-tdrg 24093 |
This theorem is referenced by: invrcn2 24112 |
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