Theorem istvc 24148

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 6842	. . . 4 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊))
2		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3	1, 2	eqtr4di 2790	. . 3 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹)
4	3	eleq1d 2822	. 2 ⊢ (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing))
5		df-tvc 24119	. 2 ⊢ TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing}
6	4, 5	elrab2 3651	1 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  Scalarcsca 17192  TopDRingctdrg 24113  TopModctlm 24114  TopVecctvc 24115

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-tvc 24119

This theorem is referenced by:  tvctdrg  24149  tvctlm  24153  nvctvc  24656