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Mirrors > Home > MPE Home > Th. List > tlmlmod | Structured version Visualization version GIF version |
Description: A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmlmod | ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
2 | eqid 2724 | . . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
3 | eqid 2724 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2724 | . . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊)) | |
5 | 1, 2, 3, 4 | istlm 24013 | . . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊)))) |
6 | 5 | simplbi 497 | . 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing)) |
7 | 6 | simp2d 1140 | 1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 Scalarcsca 17201 TopOpenctopn 17368 LModclmod 20698 ·sf cscaf 20699 Cn ccn 23052 ×t ctx 23388 TopMndctmd 23898 TopRingctrg 23984 TopModctlm 23986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-tlm 23990 |
This theorem is referenced by: tlmtgp 24024 tvclmod 24026 |
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