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Theorem tlmscatps 17925
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 17924 . 2  |-  ( W  e. TopMod  ->  F  e.  TopRing )
3 trgtps 17904 . 2  |-  ( F  e.  TopRing  ->  F  e.  TopSp )
42, 3syl 15 1  |-  ( W  e. TopMod  ->  F  e.  TopSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   ` cfv 5292  Scalarcsca 13258   TopSpctps 16690   TopRingctrg 17890  TopModctlm 17892
This theorem is referenced by:  cnmpt1vsca  17928  cnmpt2vsca  17929  tlmtgp  17930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903  df-tmd 17807  df-tgp 17808  df-trg 17894  df-tlm 17896
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