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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 28592 | . . 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 4207 | . . 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 4003 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
4 | 3 | sseli 3965 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 ran crn 5558 ‘cfv 6357 (class class class)co 7158 {coprab 7159 1c1 10540 + caddc 10542 · cmul 10544 -cneg 10873 2c2 11695 ↑cexp 13432 NrmCVeccnv 28363 CPreHilOLDccphlo 28591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ss 3954 df-ph 28592 |
This theorem is referenced by: phrel 28594 phnvi 28595 phop 28597 isph 28601 dipdi 28622 dipassr 28625 dipsubdir 28627 dipsubdi 28628 ajval 28640 minvecolem1 28653 minvecolem2 28654 minvecolem3 28655 minvecolem4a 28656 minvecolem4b 28657 minvecolem4 28659 minvecolem5 28660 minvecolem6 28661 minvecolem7 28662 |
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