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Mirrors > Home > MPE Home > Th. List > istvc | Structured version Visualization version GIF version |
Description: A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
istvc | ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊)) | |
2 | tlmtrg.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹) |
4 | 3 | eleq1d 2672 | . 2 ⊢ (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing)) |
5 | df-tvc 21776 | . 2 ⊢ TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing} | |
6 | 4, 5 | elrab2 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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