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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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 23774 | . . 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 1141 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp1d 1138 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 ↦ cmpt 5146 “ cima 5558 ‘cfv 6355 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 [,)cico 12741 √csqrt 14592 Basecbs 16483 ↾s cress 16484 Scalarcsca 16568 ·𝑖cip 16570 ℂfldccnfld 20545 PreHilcphl 20768 normcnm 23186 NrmModcnlm 23190 ℂPreHilccph 23770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-ov 7159 df-cph 23772 |
This theorem is referenced by: cphlvec 23779 cphcjcl 23787 cphipcl 23795 cphnmf 23799 cphipcj 23803 cphorthcom 23805 cphip0l 23806 cphip0r 23807 cphipeq0 23808 cphdir 23809 cphdi 23810 cph2di 23811 cphsubdir 23812 cphsubdi 23813 cph2subdi 23814 cphass 23815 cphassr 23816 ipcau 23841 nmparlem 23842 ipcn 23849 cphsscph 23854 hlphl 23968 cmscsscms 23976 bncssbn 23977 pjthlem2 24041 |
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