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Definition df-ph 27052
 Description: Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is 𝑔, the scalar product is 𝑠, and the norm is 𝑛. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
df-ph CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
Distinct variable group:   𝑔,𝑛,𝑠,𝑥,𝑦

Detailed syntax breakdown of Definition df-ph
StepHypRef Expression
1 ccphlo 27051 . 2 class CPreHilOLD
2 cnv 26823 . . 3 class NrmCVec
3 vx . . . . . . . . . . . 12 setvar 𝑥
43cv 1474 . . . . . . . . . . 11 class 𝑥
5 vy . . . . . . . . . . . 12 setvar 𝑦
65cv 1474 . . . . . . . . . . 11 class 𝑦
7 vg . . . . . . . . . . . 12 setvar 𝑔
87cv 1474 . . . . . . . . . . 11 class 𝑔
94, 6, 8co 6549 . . . . . . . . . 10 class (𝑥𝑔𝑦)
10 vn . . . . . . . . . . 11 setvar 𝑛
1110cv 1474 . . . . . . . . . 10 class 𝑛
129, 11cfv 5804 . . . . . . . . 9 class (𝑛‘(𝑥𝑔𝑦))
13 c2 10947 . . . . . . . . 9 class 2
14 cexp 12722 . . . . . . . . 9 class
1512, 13, 14co 6549 . . . . . . . 8 class ((𝑛‘(𝑥𝑔𝑦))↑2)
16 c1 9816 . . . . . . . . . . . . 13 class 1
1716cneg 10146 . . . . . . . . . . . 12 class -1
18 vs . . . . . . . . . . . . 13 setvar 𝑠
1918cv 1474 . . . . . . . . . . . 12 class 𝑠
2017, 6, 19co 6549 . . . . . . . . . . 11 class (-1𝑠𝑦)
214, 20, 8co 6549 . . . . . . . . . 10 class (𝑥𝑔(-1𝑠𝑦))
2221, 11cfv 5804 . . . . . . . . 9 class (𝑛‘(𝑥𝑔(-1𝑠𝑦)))
2322, 13, 14co 6549 . . . . . . . 8 class ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)
24 caddc 9818 . . . . . . . 8 class +
2515, 23, 24co 6549 . . . . . . 7 class (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2))
264, 11cfv 5804 . . . . . . . . . 10 class (𝑛𝑥)
2726, 13, 14co 6549 . . . . . . . . 9 class ((𝑛𝑥)↑2)
286, 11cfv 5804 . . . . . . . . . 10 class (𝑛𝑦)
2928, 13, 14co 6549 . . . . . . . . 9 class ((𝑛𝑦)↑2)
3027, 29, 24co 6549 . . . . . . . 8 class (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))
31 cmul 9820 . . . . . . . 8 class ·
3213, 30, 31co 6549 . . . . . . 7 class (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))
3325, 32wceq 1475 . . . . . 6 wff (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))
348crn 5039 . . . . . 6 class ran 𝑔
3533, 5, 34wral 2896 . . . . 5 wff 𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))
3635, 3, 34wral 2896 . . . 4 wff 𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))
3736, 7, 18, 10coprab 6550 . . 3 class {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}
382, 37cin 3539 . 2 class (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
391, 38wceq 1475 1 wff CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
 Colors of variables: wff setvar class This definition is referenced by:  phnv  27053  isphg  27056
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