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Definition df-phl 20380
Description: Define the class of all pre-Hilbert spaces (inner product spaces) over arbitrary fields with involution. (Some textbook definitions are more restrictive and require the field of scalars to be the field of real or complex numbers). (Contributed by NM, 22-Sep-2011.)
Assertion
Ref Expression
df-phl PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}
Distinct variable group:   𝑓,𝑔,,𝑣,𝑥,𝑦

Detailed syntax breakdown of Definition df-phl
StepHypRef Expression
1 cphl 20378 . 2 class PreHil
2 vf . . . . . . . . 9 setvar 𝑓
32cv 1600 . . . . . . . 8 class 𝑓
4 csr 19247 . . . . . . . 8 class *-Ring
53, 4wcel 2107 . . . . . . 7 wff 𝑓 ∈ *-Ring
6 vy . . . . . . . . . . 11 setvar 𝑦
7 vv . . . . . . . . . . . 12 setvar 𝑣
87cv 1600 . . . . . . . . . . 11 class 𝑣
96cv 1600 . . . . . . . . . . . 12 class 𝑦
10 vx . . . . . . . . . . . . 13 setvar 𝑥
1110cv 1600 . . . . . . . . . . . 12 class 𝑥
12 vh . . . . . . . . . . . . 13 setvar
1312cv 1600 . . . . . . . . . . . 12 class
149, 11, 13co 6924 . . . . . . . . . . 11 class (𝑦𝑥)
156, 8, 14cmpt 4967 . . . . . . . . . 10 class (𝑦𝑣 ↦ (𝑦𝑥))
16 vg . . . . . . . . . . . 12 setvar 𝑔
1716cv 1600 . . . . . . . . . . 11 class 𝑔
18 crglmod 19577 . . . . . . . . . . . 12 class ringLMod
193, 18cfv 6137 . . . . . . . . . . 11 class (ringLMod‘𝑓)
20 clmhm 19425 . . . . . . . . . . 11 class LMHom
2117, 19, 20co 6924 . . . . . . . . . 10 class (𝑔 LMHom (ringLMod‘𝑓))
2215, 21wcel 2107 . . . . . . . . 9 wff (𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓))
2311, 11, 13co 6924 . . . . . . . . . . 11 class (𝑥𝑥)
24 c0g 16497 . . . . . . . . . . . 12 class 0g
253, 24cfv 6137 . . . . . . . . . . 11 class (0g𝑓)
2623, 25wceq 1601 . . . . . . . . . 10 wff (𝑥𝑥) = (0g𝑓)
2717, 24cfv 6137 . . . . . . . . . . 11 class (0g𝑔)
2811, 27wceq 1601 . . . . . . . . . 10 wff 𝑥 = (0g𝑔)
2926, 28wi 4 . . . . . . . . 9 wff ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔))
3011, 9, 13co 6924 . . . . . . . . . . . 12 class (𝑥𝑦)
31 cstv 16351 . . . . . . . . . . . . 13 class *𝑟
323, 31cfv 6137 . . . . . . . . . . . 12 class (*𝑟𝑓)
3330, 32cfv 6137 . . . . . . . . . . 11 class ((*𝑟𝑓)‘(𝑥𝑦))
3433, 14wceq 1601 . . . . . . . . . 10 wff ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)
3534, 6, 8wral 3090 . . . . . . . . 9 wff 𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)
3622, 29, 35w3a 1071 . . . . . . . 8 wff ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))
3736, 10, 8wral 3090 . . . . . . 7 wff 𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))
385, 37wa 386 . . . . . 6 wff (𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))
39 csca 16352 . . . . . . 7 class Scalar
4017, 39cfv 6137 . . . . . 6 class (Scalar‘𝑔)
4138, 2, 40wsbc 3652 . . . . 5 wff [(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))
42 cip 16354 . . . . . 6 class ·𝑖
4317, 42cfv 6137 . . . . 5 class (·𝑖𝑔)
4441, 12, 43wsbc 3652 . . . 4 wff [(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))
45 cbs 16266 . . . . 5 class Base
4617, 45cfv 6137 . . . 4 class (Base‘𝑔)
4744, 7, 46wsbc 3652 . . 3 wff [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))
48 clvec 19508 . . 3 class LVec
4947, 16, 48crab 3094 . 2 class {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}
501, 49wceq 1601 1 wff PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}
Colors of variables: wff setvar class
This definition is referenced by:  isphl  20382
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