![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tvclmod | Structured version Visualization version GIF version |
Description: A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tvclmod | ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tvctlm 22511 | . 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod) | |
2 | tlmlmod 22503 | . 2 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 LModclmod 19359 TopModctlm 22472 TopVecctvc 22473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-iota 6154 df-fv 6198 df-ov 6981 df-tlm 22476 df-tvc 22477 |
This theorem is referenced by: tvclvec 22513 |
Copyright terms: Public domain | W3C validator |