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Theorem tlmtmd 17885
Description: A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tlmtmd  |-  ( W  e. TopMod  ->  W  e. TopMnd )

Proof of Theorem tlmtmd
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( .s f `  W )  =  ( .s f `  W )
2 eqid 2296 . . . 4  |-  ( TopOpen `  W )  =  (
TopOpen `  W )
3 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2296 . . . 4  |-  ( TopOpen `  (Scalar `  W ) )  =  ( TopOpen `  (Scalar `  W ) )
51, 2, 3, 4istlm 17883 . . 3  |-  ( W  e. TopMod 
<->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  (Scalar `  W )  e.  TopRing )  /\  ( .s f `  W )  e.  ( ( (
TopOpen `  (Scalar `  W
) )  tX  ( TopOpen
`  W ) )  Cn  ( TopOpen `  W
) ) ) )
65simplbi 446 . 2  |-  ( W  e. TopMod  ->  ( W  e. TopMnd  /\  W  e.  LMod  /\  (Scalar `  W )  e.  TopRing ) )
76simp1d 967 1  |-  ( W  e. TopMod  ->  W  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1696   ` cfv 5271  (class class class)co 5874  Scalarcsca 13227   TopOpenctopn 13342   LModclmod 15643   .s fcscaf 15644    Cn ccn 16970    tX ctx 17271  TopMndctmd 17769   TopRingctrg 17854  TopModctlm 17856
This theorem is referenced by:  tlmtps  17886  tlmtgp  17894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-tlm 17860
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