Theorem tlmtmd 24416

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 2760	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2760	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2760	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2760	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 24414	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 502	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp1d 1160	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2145  ‘cfv 6533  (class class class)co 7414  Scalarcsca 17348  TopOpenctopn 17509  LModclmod 21047   ·_sf cscaf 21048   Cn ccn 23452   ×_t ctx 23789  TopMndctmd 24299  TopRingctrg 24385  TopModctlm 24387

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-tlm 24391

This theorem is used by:  tlmtps  24417  tlmtgp  24425