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Mirrors > Home > MPE Home > Th. List > phllvec | Structured version Visualization version GIF version |
Description: A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phllvec | ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2726 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2726 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2726 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2726 | . . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊)) | |
6 | eqid 2726 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21624 | . 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (Scalar‘𝑊) ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))) |
8 | 7 | simp1bi 1142 | 1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 *𝑟cstv 17268 Scalarcsca 17269 ·𝑖cip 17271 0gc0g 17454 *-Ringcsr 20817 LMHom clmhm 20997 LVecclvec 21080 ringLModcrglmod 21150 PreHilcphl 21620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-iota 6506 df-fv 6562 df-ov 7427 df-phl 21622 |
This theorem is referenced by: phllmod 21626 phlssphl 21655 obsne0 21723 obslbs 21728 cphlvec 25194 phclm 25251 ipcau2 25253 tcphcph 25256 |
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