| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cphphl | Structured version Visualization version GIF version | ||
| Description: A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphphl | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) |
| 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 | simp1d 1143 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) |
| 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: cphlvec 25209 cphcjcl 25217 cphipcl 25225 cphnmf 25229 cphipcj 25233 cphorthcom 25235 cphip0l 25236 cphip0r 25237 cphipeq0 25238 cphdir 25239 cphdi 25240 cph2di 25241 cphsubdir 25242 cphsubdi 25243 cph2subdi 25244 cphass 25245 cphassr 25246 ipcau 25272 nmparlem 25273 ipcn 25280 cphsscph 25285 hlphl 25399 cmscsscms 25407 bncssbn 25408 pjthlem2 25472 |
| Copyright terms: Public domain | W3C validator |