MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tlmscatps Structured version   Unicode version

Theorem tlmscatps 18220
Description: The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
tlmscatps  |-  ( W  e. TopMod  ->  F  e.  TopSp )

Proof of Theorem tlmscatps
StepHypRef Expression
1 tlmtrg.f . . 3  |-  F  =  (Scalar `  W )
21tlmtrg 18219 . 2  |-  ( W  e. TopMod  ->  F  e.  TopRing )
3 trgtps 18199 . 2  |-  ( F  e.  TopRing  ->  F  e.  TopSp )
42, 3syl 16 1  |-  ( W  e. TopMod  ->  F  e.  TopSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  Scalarcsca 13532   TopSpctps 16961   TopRingctrg 18185  TopModctlm 18187
This theorem is referenced by:  cnmpt1vsca  18223  cnmpt2vsca  18224  tlmtgp  18225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-tmd 18102  df-tgp 18103  df-trg 18189  df-tlm 18191
  Copyright terms: Public domain W3C validator