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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 27668 | . . 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 3833 | . . 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 3635 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
4 | 3 | sseli 3599 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ran crn 5115 ‘cfv 5888 (class class class)co 6650 {coprab 6651 1c1 9937 + caddc 9939 · cmul 9941 -cneg 10267 2c2 11070 ↑cexp 12860 NrmCVeccnv 27439 CPreHilOLDccphlo 27667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-ph 27668 |
This theorem is referenced by: phrel 27670 phnvi 27671 phop 27673 isph 27677 dipdi 27698 dipassr 27701 dipsubdir 27703 dipsubdi 27704 sspph 27710 ajval 27717 minvecolem1 27730 minvecolem2 27731 minvecolem3 27732 minvecolem4a 27733 minvecolem4b 27734 minvecolem4 27736 minvecolem5 27737 minvecolem6 27738 minvecolem7 27739 |
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