Theorem tlmtmd 24301

Description: A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
tlmtmd	⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Proof of Theorem tlmtmd

Step	Hyp	Ref	Expression
1		eqid 2765	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2765	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2765	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2765	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 24299	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 501	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp1d 1158	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Syntax hints:   → wi 4   ∧ w3a 1101   ∈ wcel 2145  ‘cfv 6525  (class class class)co 7400  Scalarcsca 17301  TopOpenctopn 17462  LModclmod 20947   ·_sf cscaf 20948   Cn ccn 23338   ×_t ctx 23674  TopMndctmd 24184  TopRingctrg 24270  TopModctlm 24272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ov 7403  df-tlm 24276

This theorem is referenced by:  tlmtps  24302  tlmtgp  24310