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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 2777 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2777 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2777 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2777 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2777 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 23377 | . . 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 1136 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp2d 1134 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∩ cin 3790 ⊆ wss 3791 ↦ cmpt 4965 “ cima 5358 ‘cfv 6135 (class class class)co 6922 0cc0 10272 +∞cpnf 10408 [,)cico 12489 √csqrt 14380 Basecbs 16255 ↾s cress 16256 Scalarcsca 16341 ·𝑖cip 16343 ℂfldccnfld 20142 PreHilcphl 20367 normcnm 22789 NrmModcnlm 22793 ℂPreHilccph 23373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-xp 5361 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fv 6143 df-ov 6925 df-cph 23375 |
This theorem is referenced by: cphngp 23380 cphlmod 23381 cphnvc 23383 cphnmvs 23397 ipcnlem2 23450 ipcnlem1 23451 csscld 23455 |
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