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Mirrors > Home > MPE Home > Th. List > cphnlm | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
cphnlm | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2771 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2771 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2771 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2771 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 23189 | . . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥))))) |
7 | 6 | simp1bi 1139 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp2d 1137 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ⊆ wss 3723 ↦ cmpt 4864 “ cima 5253 ‘cfv 6030 (class class class)co 6796 0cc0 10142 +∞cpnf 10277 [,)cico 12382 √csqrt 14181 Basecbs 16064 ↾s cress 16065 Scalarcsca 16152 ·𝑖cip 16154 ℂfldccnfld 19961 PreHilcphl 20186 normcnm 22601 NrmModcnlm 22605 ℂPreHilccph 23185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-nul 4924 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-xp 5256 df-cnv 5258 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fv 6038 df-ov 6799 df-cph 23187 |
This theorem is referenced by: cphngp 23192 cphlmod 23193 cphnvc 23195 cphnmvs 23209 ipcnlem2 23262 ipcnlem1 23263 csscld 23267 |
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