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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 28596 | . . 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 4155 | . . 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 3949 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
4 | 3 | sseli 3911 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 ran crn 5520 ‘cfv 6324 (class class class)co 7135 {coprab 7136 1c1 10527 + caddc 10529 · cmul 10531 -cneg 10860 2c2 11680 ↑cexp 13425 NrmCVeccnv 28367 CPreHilOLDccphlo 28595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-ph 28596 |
This theorem is referenced by: phrel 28598 phnvi 28599 phop 28601 isph 28605 dipdi 28626 dipassr 28629 dipsubdir 28631 dipsubdi 28632 ajval 28644 minvecolem1 28657 minvecolem2 28658 minvecolem3 28659 minvecolem4a 28660 minvecolem4b 28661 minvecolem4 28663 minvecolem5 28664 minvecolem6 28665 minvecolem7 28666 |
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