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| 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 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
| 3 | eqid 2737 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 5 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 6 | 1, 2, 3, 4, 5 | iscph 25204 | . . 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 1146 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) | 
| 8 | 7 | simp2d 1144 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ↦ cmpt 5225 “ cima 5688 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,)cico 13389 √csqrt 15272 Basecbs 17247 ↾s cress 17274 Scalarcsca 17300 ·𝑖cip 17302 ℂfldccnfld 21364 PreHilcphl 21642 normcnm 24589 NrmModcnlm 24593 ℂPreHilccph 25200 | 
| 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-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 df-cph 25202 | 
| This theorem is referenced by: cphngp 25207 cphlmod 25208 cphnvc 25210 cphnmvs 25224 ipcnlem2 25278 ipcnlem1 25279 csscld 25283 cphsscph 25285 | 
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