Theorem tlmtmd 24216

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 2740	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2740	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2740	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2740	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 24214	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 497	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp1d 1142	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Scalarcsca 17314  TopOpenctopn 17481  LModclmod 20880   ·_sf cscaf 20881   Cn ccn 23253   ×_t ctx 23589  TopMndctmd 24099  TopRingctrg 24185  TopModctlm 24187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-tlm 24191

This theorem is referenced by:  tlmtps  24217  tlmtgp  24225