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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 24193 | . 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))) |
| 6 | 5 | simprbi 496 | 1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Scalarcsca 17300 TopOpenctopn 17466 LModclmod 20858 ·sf cscaf 20859 Cn ccn 23232 ×t ctx 23568 TopMndctmd 24078 TopRingctrg 24164 TopModctlm 24166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tlm 24170 |
| This theorem is referenced by: cnmpt1vsca 24202 cnmpt2vsca 24203 |
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