Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2824 | . . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
2 | eqid 2824 | . . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
3 | tlmtrg.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | eqid 2824 | . . . 4 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
5 | 1, 2, 3, 4 | istlm 22796 | . . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘𝐹) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊)))) |
6 | 5 | simplbi 500 | . 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing)) |
7 | 6 | simp3d 1140 | 1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Scalarcsca 16571 TopOpenctopn 16698 LModclmod 19637 ·sf cscaf 19638 Cn ccn 21835 ×t ctx 22171 TopMndctmd 22681 TopRingctrg 22767 TopModctlm 22769 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-tlm 22773 |
This theorem is referenced by: tlmscatps 22802 |
Copyright terms: Public domain | W3C validator |