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Mirrors > Home > MPE Home > Th. List > cphsca | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a vector space over a subfield of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphsca.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphsca | ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2823 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2823 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | cphsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | cphsca.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
6 | 1, 2, 3, 4, 5 | iscph 23776 | . . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥))))) |
7 | 6 | simp1bi 1141 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾))) |
8 | 7 | simp3d 1140 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ↦ cmpt 5148 “ cima 5560 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 [,)cico 12743 √csqrt 14594 Basecbs 16485 ↾s cress 16486 Scalarcsca 16570 ·𝑖cip 16572 ℂfldccnfld 20547 PreHilcphl 20770 normcnm 23188 NrmModcnlm 23192 ℂPreHilccph 23772 |
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 2795 ax-nul 5212 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-ov 7161 df-cph 23774 |
This theorem is referenced by: cphsubrg 23786 cphreccl 23787 cphcjcl 23789 cphqss 23794 cphclm 23795 ipcau 23843 cphsscph 23856 hlprlem 23972 ishl2 23975 |
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