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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 2798 | . . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
2 | eqid 2798 | . . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
3 | eqid 2798 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2798 | . . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊)) | |
5 | 1, 2, 3, 4 | istlm 22790 | . . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊)))) |
6 | 5 | simplbi 501 | . 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 2111 ‘cfv 6324 (class class class)co 7135 Scalarcsca 16560 TopOpenctopn 16687 LModclmod 19627 ·sf cscaf 19628 Cn ccn 21829 ×t ctx 22165 TopMndctmd 22675 TopRingctrg 22761 TopModctlm 22763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-tlm 22767 |
This theorem is referenced by: tlmtps 22793 tlmtgp 22801 |
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