Theorem istvc 23324

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.

Step	Hyp	Ref	Expression
1		fveq2 6768	. . . 4 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3	1, 2	eqtr4di 2797	. . 3 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
4	3	eleq1d 2824	. 2 ⊢ (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5		df-tvc 23295	. 2 ⊢ TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
6	4, 5	elrab2 3628	1 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  Scalarcsca 16946  TopDRingctdrg 23289  TopModctlm 23290  TopVecctvc 23291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-tvc 23295

This theorem is referenced by:  tvctdrg  23325  tvctlm  23329  nvctvc  23845