Theorem istvc 23696

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 6892	. . . 4 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3	1, 2	eqtr4di 2791	. . 3 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
4	3	eleq1d 2819	. 2 ⊢ (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5		df-tvc 23667	. 2 ⊢ TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
6	4, 5	elrab2 3687	1 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ‘cfv 6544  Scalarcsca 17200  TopDRingctdrg 23661  TopModctlm 23662  TopVecctvc 23663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-tvc 23667

This theorem is referenced by:  tvctdrg  23697  tvctlm  23701  nvctvc  24217