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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 2733 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2733 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2733 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2733 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2733 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 24687 | . . 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 1088 = wceq 1542 ∈ wcel 2107 ∩ cin 3948 ⊆ wss 3949 ↦ cmpt 5232 “ cima 5680 ‘cfv 6544 (class class class)co 7409 0cc0 11110 +∞cpnf 11245 [,)cico 13326 √csqrt 15180 Basecbs 17144 ↾s cress 17173 Scalarcsca 17200 ·𝑖cip 17202 ℂfldccnfld 20944 PreHilcphl 21177 normcnm 24085 NrmModcnlm 24089 ℂPreHilccph 24683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 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 7412 df-cph 24685 |
This theorem is referenced by: cphlvec 24692 cphcjcl 24700 cphipcl 24708 cphnmf 24712 cphipcj 24716 cphorthcom 24718 cphip0l 24719 cphip0r 24720 cphipeq0 24721 cphdir 24722 cphdi 24723 cph2di 24724 cphsubdir 24725 cphsubdi 24726 cph2subdi 24727 cphass 24728 cphassr 24729 ipcau 24755 nmparlem 24756 ipcn 24763 cphsscph 24768 hlphl 24882 cmscsscms 24890 bncssbn 24891 pjthlem2 24955 |
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