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Mirrors > Home > MPE Home > Th. List > vscacn | Structured version Visualization version GIF version |
Description: The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istlm.s | ⊢ · = ( ·sf ‘𝑊) |
istlm.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
istlm.f | ⊢ 𝐹 = (Scalar‘𝑊) |
istlm.k | ⊢ 𝐾 = (TopOpen‘𝐹) |
Ref | Expression |
---|---|
vscacn | ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istlm.s | . . 3 ⊢ · = ( ·sf ‘𝑊) | |
2 | istlm.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑊) | |
3 | istlm.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | istlm.k | . . 3 ⊢ 𝐾 = (TopOpen‘𝐹) | |
5 | 1, 2, 3, 4 | istlm 22787 | . 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))) |
6 | 5 | simprbi 499 | 1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Scalarcsca 16562 TopOpenctopn 16689 LModclmod 19628 ·sf cscaf 19629 Cn ccn 21826 ×t ctx 22162 TopMndctmd 22672 TopRingctrg 22758 TopModctlm 22760 |
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 |
This theorem is referenced by: cnmpt1vsca 22796 cnmpt2vsca 22797 |
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