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Mirrors > Home > MPE Home > Th. List > tlmtrg | Structured version Visualization version GIF version |
Description: The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
tlmtrg | ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
2 | eqid 2610 | . . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
3 | tlmtrg.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | eqid 2610 | . . . 4 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
5 | 1, 2, 3, 4 | istlm 21798 | . . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘𝐹) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊)))) |
6 | 5 | simplbi 475 | . 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing)) |
7 | 6 | simp3d 1068 | 1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Scalarcsca 15771 TopOpenctopn 15905 LModclmod 18686 ·sf cscaf 18687 Cn ccn 20838 ×t ctx 21173 TopMndctmd 21684 TopRingctrg 21769 TopModctlm 21771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-tlm 21775 |
This theorem is referenced by: tlmscatps 21804 |
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