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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 28517 | . . 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 4202 | . . 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 3998 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
4 | 3 | sseli 3960 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ran crn 5549 ‘cfv 6348 (class class class)co 7145 {coprab 7146 1c1 10526 + caddc 10528 · cmul 10530 -cneg 10859 2c2 11680 ↑cexp 13417 NrmCVeccnv 28288 CPreHilOLDccphlo 28516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-ph 28517 |
This theorem is referenced by: phrel 28519 phnvi 28520 phop 28522 isph 28526 dipdi 28547 dipassr 28550 dipsubdir 28552 dipsubdi 28553 ajval 28565 minvecolem1 28578 minvecolem2 28579 minvecolem3 28580 minvecolem4a 28581 minvecolem4b 28582 minvecolem4 28584 minvecolem5 28585 minvecolem6 28586 minvecolem7 28587 |
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