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Mirrors > Home > MPE Home > Th. List > tlmscatps | Structured version Visualization version GIF version |
Description: The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
tlmscatps | ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlmtrg.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | tlmtrg 21993 | . 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) |
3 | trgtps 21973 | . 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 Scalarcsca 15944 TopSpctps 20736 TopRingctrg 21959 TopModctlm 21961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-tmd 21876 df-tgp 21877 df-trg 21963 df-tlm 21965 |
This theorem is referenced by: cnmpt1vsca 21997 cnmpt2vsca 21998 tlmtgp 21999 |
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