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Mirrors > Home > MPE Home > Th. List > tlmtmd | Structured version Visualization version GIF version |
Description: A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtmd | ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
2 | eqid 2726 | . . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
3 | eqid 2726 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2726 | . . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊)) | |
5 | 1, 2, 3, 4 | istlm 24040 | . . 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 | simp1d 1139 | 1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 Scalarcsca 17207 TopOpenctopn 17374 LModclmod 20704 ·sf cscaf 20705 Cn ccn 23079 ×t ctx 23415 TopMndctmd 23925 TopRingctrg 24011 TopModctlm 24013 |
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 2697 |
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 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-tlm 24017 |
This theorem is referenced by: tlmtps 24043 tlmtgp 24051 |
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