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| 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 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
| 4 | eqid 2737 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 5 | eqid 2737 | . . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊)) | |
| 6 | eqid 2737 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
| 7 | 1, 2, 3, 4, 5, 6 | isphl 21646 | . 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 1146 | 1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 *𝑟cstv 17299 Scalarcsca 17300 ·𝑖cip 17302 0gc0g 17484 *-Ringcsr 20839 LMHom clmhm 21018 LVecclvec 21101 ringLModcrglmod 21171 PreHilcphl 21642 | 
| 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 ax-nul 5306 | 
| 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-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 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-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-ov 7434 df-phl 21644 | 
| This theorem is referenced by: phllmod 21648 phlssphl 21677 obsne0 21745 obslbs 21750 cphlvec 25209 phclm 25266 ipcau2 25268 tcphcph 25271 | 
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