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Theorem istvc 24216
Description: A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
istvc (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))

Proof of Theorem istvc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2 tlmtrg.f . . . 4 𝐹 = (Scalar‘𝑊)
31, 2eqtr4di 2793 . . 3 (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
43eleq1d 2824 . 2 (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5 df-tvc 24187 . 2 TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
64, 5elrab2 3698 1 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  Scalarcsca 17301  TopDRingctdrg 24181  TopModctlm 24182  TopVecctvc 24183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-tvc 24187
This theorem is referenced by:  tvctdrg  24217  tvctlm  24221  nvctvc  24737
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