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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 6664 | . . . 4 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = (Scalar‘𝑊)) | |
2 | tlmtrg.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑥 = 𝑊 → (Scalar‘𝑥) = 𝐹) |
4 | 3 | eleq1d 2897 | . 2 ⊢ (𝑥 = 𝑊 → ((Scalar‘𝑥) ∈ TopDRing ↔ 𝐹 ∈ TopDRing)) |
5 | df-tvc 22765 | . 2 ⊢ TopVec = {𝑥 ∈ TopMod ∣ (Scalar‘𝑥) ∈ TopDRing} | |
6 | 4, 5 | elrab2 3682 | 1 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 Scalarcsca 16562 TopDRingctdrg 22759 TopModctlm 22760 TopVecctvc 22761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-tvc 22765 |
This theorem is referenced by: tvctdrg 22795 tvctlm 22799 nvctvc 23303 |
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