Theorem tlmtps 24166

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

Assertion

Ref	Expression
tlmtps	⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Proof of Theorem tlmtps

Step	Hyp	Ref	Expression
1		tlmtmd 24165	. 2 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
2		tmdtps 24054	. 2 ⊢ (𝑊 ∈ TopMnd → 𝑊 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2114  TopSpctps 22910  TopMndctmd 24048  TopModctlm 24136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364  df-tmd 24050  df-tlm 24140

This theorem is referenced by:  cnmpt1vsca  24172  cnmpt2vsca  24173  tlmtgp  24174