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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 2730 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2730 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2730 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2730 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2730 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 24920 | . . 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 1143 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp1d 1140 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∩ cin 3948 ⊆ wss 3949 ↦ cmpt 5232 “ cima 5680 ‘cfv 6544 (class class class)co 7413 0cc0 11114 +∞cpnf 11251 [,)cico 13332 √csqrt 15186 Basecbs 17150 ↾s cress 17179 Scalarcsca 17206 ·𝑖cip 17208 ℂfldccnfld 21146 PreHilcphl 21398 normcnm 24307 NrmModcnlm 24311 ℂPreHilccph 24916 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fv 6552 df-ov 7416 df-cph 24918 |
This theorem is referenced by: cphlvec 24925 cphcjcl 24933 cphipcl 24941 cphnmf 24945 cphipcj 24949 cphorthcom 24951 cphip0l 24952 cphip0r 24953 cphipeq0 24954 cphdir 24955 cphdi 24956 cph2di 24957 cphsubdir 24958 cphsubdi 24959 cph2subdi 24960 cphass 24961 cphassr 24962 ipcau 24988 nmparlem 24989 ipcn 24996 cphsscph 25001 hlphl 25115 cmscsscms 25123 bncssbn 25124 pjthlem2 25188 |
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