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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 2739 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2739 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2739 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2739 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2739 | . . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊)) | |
6 | eqid 2739 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20622 | . 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 1147 | 1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3064 ↦ cmpt 5151 ‘cfv 6400 (class class class)co 7234 Basecbs 16792 *𝑟cstv 16836 Scalarcsca 16837 ·𝑖cip 16839 0gc0g 16976 *-Ringcsr 19912 LMHom clmhm 20088 LVecclvec 20171 ringLModcrglmod 20238 PreHilcphl 20618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-nul 5215 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-mpt 5152 df-iota 6358 df-fv 6408 df-ov 7237 df-phl 20620 |
This theorem is referenced by: phllmod 20624 phlssphl 20653 obsne0 20719 obslbs 20724 cphlvec 24103 phclm 24160 ipcau2 24162 tcphcph 24165 |
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