![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23533 | . 2 β’ (π β TopDRing β (π β TopRing β§ π β DivRing β§ (π βΎs π) β TopGrp)) |
4 | 3 | simp3bi 1148 | 1 β’ (π β TopDRing β (π βΎs π) β TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 βΎs cress 17117 mulGrpcmgp 19901 Unitcui 20073 DivRingcdr 20197 TopGrpctgp 23438 TopRingctrg 23523 TopDRingctdrg 23524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-tdrg 23528 |
This theorem is referenced by: invrcn2 23547 |
Copyright terms: Public domain | W3C validator |