![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tgpcn | Structured version Visualization version GIF version |
Description: In a topological group, the operation πΉ representing the functionalization of the operator slot +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | β’ π½ = (TopOpenβπΊ) |
tgpcn.1 | β’ πΉ = (+πβπΊ) |
Ref | Expression |
---|---|
tgpcn | β’ (πΊ β TopGrp β πΉ β ((π½ Γt π½) Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptmd 23582 | . 2 β’ (πΊ β TopGrp β πΊ β TopMnd) | |
2 | tgpcn.j | . . 3 β’ π½ = (TopOpenβπΊ) | |
3 | tgpcn.1 | . . 3 β’ πΉ = (+πβπΊ) | |
4 | 2, 3 | tmdcn 23586 | . 2 β’ (πΊ β TopMnd β πΉ β ((π½ Γt π½) Cn π½)) |
5 | 1, 4 | syl 17 | 1 β’ (πΊ β TopGrp β πΉ β ((π½ Γt π½) Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7408 TopOpenctopn 17366 +πcplusf 18557 Cn ccn 22727 Γt ctx 23063 TopMndctmd 23573 TopGrpctgp 23574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-tmd 23575 df-tgp 23576 |
This theorem is referenced by: pl1cn 32930 |
Copyright terms: Public domain | W3C validator |