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Theorem istvc 23687
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
istvc (π‘Š ∈ TopVec ↔ (π‘Š ∈ TopMod ∧ 𝐹 ∈ TopDRing))

Proof of Theorem istvc
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6888 . . . 4 (π‘₯ = π‘Š β†’ (Scalarβ€˜π‘₯) = (Scalarβ€˜π‘Š))
2 tlmtrg.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
31, 2eqtr4di 2790 . . 3 (π‘₯ = π‘Š β†’ (Scalarβ€˜π‘₯) = 𝐹)
43eleq1d 2818 . 2 (π‘₯ = π‘Š β†’ ((Scalarβ€˜π‘₯) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5 df-tvc 23658 . 2 TopVec = {π‘₯ ∈ TopMod ∣ (Scalarβ€˜π‘₯) ∈ TopDRing}
64, 5elrab2 3685 1 (π‘Š ∈ TopVec ↔ (π‘Š ∈ TopMod ∧ 𝐹 ∈ TopDRing))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  β€˜cfv 6540  Scalarcsca 17196  TopDRingctdrg 23652  TopModctlm 23653  TopVecctvc 23654
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-tvc 23658
This theorem is referenced by:  tvctdrg  23688  tvctlm  23692  nvctvc  24208
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