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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 2739 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2739 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2739 | . . . 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 24315 | . . 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 1143 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾))) |
8 | 7 | simp3d 1142 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∩ cin 3890 ⊆ wss 3891 ↦ cmpt 5161 “ cima 5591 ‘cfv 6430 (class class class)co 7268 0cc0 10855 +∞cpnf 10990 [,)cico 13063 √csqrt 14925 Basecbs 16893 ↾s cress 16922 Scalarcsca 16946 ·𝑖cip 16948 ℂfldccnfld 20578 PreHilcphl 20810 normcnm 23713 NrmModcnlm 23717 ℂPreHilccph 24311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-nul 5233 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-xp 5594 df-cnv 5596 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fv 6438 df-ov 7271 df-cph 24313 |
This theorem is referenced by: cphsubrg 24325 cphreccl 24326 cphcjcl 24328 cphqss 24333 cphclm 24334 ipcau 24383 cphsscph 24396 hlprlem 24512 ishl2 24515 |
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