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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 2735 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2735 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2735 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2735 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2735 | . . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊)) | |
6 | eqid 2735 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21664 | . 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 1144 | 1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 *𝑟cstv 17300 Scalarcsca 17301 ·𝑖cip 17303 0gc0g 17486 *-Ringcsr 20856 LMHom clmhm 21036 LVecclvec 21119 ringLModcrglmod 21189 PreHilcphl 21660 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-phl 21662 |
This theorem is referenced by: phllmod 21666 phlssphl 21695 obsne0 21763 obslbs 21768 cphlvec 25223 phclm 25280 ipcau2 25282 tcphcph 25285 |
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