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Theorem istvc 21805
 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 6103 . . . 4 (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2 tlmtrg.f . . . 4 𝐹 = (Scalar‘𝑊)
31, 2syl6eqr 2662 . . 3 (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
43eleq1d 2672 . 2 (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5 df-tvc 21776 . 2 TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
64, 5elrab2 3333 1 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Scalarcsca 15771  TopDRingctdrg 21770  TopModctlm 21771  TopVecctvc 21772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-tvc 21776 This theorem is referenced by:  tvctdrg  21806  tvctlm  21810  nvctvc  22314
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