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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 22799 | . 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod) | |
2 | tlmlmod 22791 | . 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 2110 LModclmod 19628 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 2156 ax-12 2172 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 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-ov 7153 df-tlm 22764 df-tvc 22765 |
This theorem is referenced by: tvclvec 22801 |
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