Theorem tlmtrg 21993

Description: The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
tlmtrg.f	|- F = (Scalar ` W )

Assertion

Ref	Expression
tlmtrg	|- ( W e. TopMod -> F e. TopRing )

Proof of Theorem tlmtrg

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( .sf ` W ) = ( .sf ` W )
2		eqid 2622	. . . 4 |- ( TopOpen ` W ) = ( TopOpen ` W )
3		tlmtrg.f	. . . 4 |- F = (Scalar ` W )
4		eqid 2622	. . . 4 |- ( TopOpen ` F ) = ( TopOpen ` F )
5	1, 2, 3, 4	istlm 21988	. . 3 |- ( W e. TopMod <-> ( ( W e. TopMnd /\ W e. LMod /\ F e. TopRing ) /\ ( .sf ` W ) e. ( ( ( TopOpen ` F ) tX ( TopOpen ` W ) ) Cn ( TopOpen ` W ) ) ) )
6	5	simplbi 476	. 2 |- ( W e. TopMod -> ( W e. TopMnd /\ W e. LMod /\ F e. TopRing ) )
7	6	simp3d 1075	1 |- ( W e. TopMod -> F e. TopRing )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650  Scalarcsca 15944   TopOpenctopn 16082   LModclmod 18863   .sfcscaf 18864   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874   TopRingctrg 21959  TopModctlm 21961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tlm 21965

This theorem is referenced by:  tlmscatps  21994