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Mirrors > Home > MPE Home > Th. List > phnv | Structured version Visualization version GIF version |
Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
phnv | ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ph 27052 | . . 3 ⊢ CPreHilOLD = (NrmCVec ∩ {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) | |
2 | inss1 3795 | . . 3 ⊢ (NrmCVec ∩ {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) ⊆ NrmCVec | |
3 | 1, 2 | eqsstri 3598 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
4 | 3 | sseli 3564 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 ran crn 5039 ‘cfv 5804 (class class class)co 6549 {coprab 6550 1c1 9816 + caddc 9818 · cmul 9820 -cneg 10146 2c2 10947 ↑cexp 12722 NrmCVeccnv 26823 CPreHilOLDccphlo 27051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-ph 27052 |
This theorem is referenced by: phrel 27054 phnvi 27055 phop 27057 isph 27061 dipdi 27082 dipassr 27085 dipsubdir 27087 dipsubdi 27088 sspph 27094 ajval 27101 minvecolem1 27114 minvecolem2 27115 minvecolem3 27116 minvecolem4a 27117 minvecolem4b 27118 minvecolem4 27120 minvecolem5 27121 minvecolem6 27122 minvecolem7 27123 |
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