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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 2726 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2726 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2726 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2726 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2726 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 25189 | . . 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 1142 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp1d 1139 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ⊆ wss 3947 ↦ cmpt 5236 “ cima 5685 ‘cfv 6554 (class class class)co 7424 0cc0 11158 +∞cpnf 11295 [,)cico 13380 √csqrt 15238 Basecbs 17213 ↾s cress 17242 Scalarcsca 17269 ·𝑖cip 17271 ℂfldccnfld 21343 PreHilcphl 21620 normcnm 24576 NrmModcnlm 24580 ℂPreHilccph 25185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fv 6562 df-ov 7427 df-cph 25187 |
This theorem is referenced by: cphlvec 25194 cphcjcl 25202 cphipcl 25210 cphnmf 25214 cphipcj 25218 cphorthcom 25220 cphip0l 25221 cphip0r 25222 cphipeq0 25223 cphdir 25224 cphdi 25225 cph2di 25226 cphsubdir 25227 cphsubdi 25228 cph2subdi 25229 cphass 25230 cphassr 25231 ipcau 25257 nmparlem 25258 ipcn 25265 cphsscph 25270 hlphl 25384 cmscsscms 25392 bncssbn 25393 pjthlem2 25457 |
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