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Mirrors > Home > MPE Home > Th. List > vscacn | Structured version Visualization version GIF version |
Description: The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istlm.s | β’ Β· = ( Β·sf βπ) |
istlm.j | β’ π½ = (TopOpenβπ) |
istlm.f | β’ πΉ = (Scalarβπ) |
istlm.k | β’ πΎ = (TopOpenβπΉ) |
Ref | Expression |
---|---|
vscacn | β’ (π β TopMod β Β· β ((πΎ Γt π½) Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istlm.s | . . 3 β’ Β· = ( Β·sf βπ) | |
2 | istlm.j | . . 3 β’ π½ = (TopOpenβπ) | |
3 | istlm.f | . . 3 β’ πΉ = (Scalarβπ) | |
4 | istlm.k | . . 3 β’ πΎ = (TopOpenβπΉ) | |
5 | 1, 2, 3, 4 | istlm 23552 | . 2 β’ (π β TopMod β ((π β TopMnd β§ π β LMod β§ πΉ β TopRing) β§ Β· β ((πΎ Γt π½) Cn π½))) |
6 | 5 | simprbi 498 | 1 β’ (π β TopMod β Β· β ((πΎ Γt π½) Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Scalarcsca 17141 TopOpenctopn 17308 LModclmod 20336 Β·sf cscaf 20337 Cn ccn 22591 Γt ctx 22927 TopMndctmd 23437 TopRingctrg 23523 TopModctlm 23525 |
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-tlm 23529 |
This theorem is referenced by: cnmpt1vsca 23561 cnmpt2vsca 23562 |
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