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Theorem istvc 24110
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 6830 . . . 4 (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2 tlmtrg.f . . . 4 𝐹 = (Scalar‘𝑊)
31, 2eqtr4di 2786 . . 3 (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
43eleq1d 2818 . 2 (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5 df-tvc 24081 . 2 TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
64, 5elrab2 3646 1 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1541  wcel 2113  cfv 6488  Scalarcsca 17168  TopDRingctdrg 24075  TopModctlm 24076  TopVecctvc 24077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-tvc 24081
This theorem is referenced by:  tvctdrg  24111  tvctlm  24115  nvctvc  24618
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