Definition df-phl 21617

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][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))}

Distinct variable group:   𝑓,𝑔,ℎ,𝑣,𝑥,𝑦

Detailed syntax breakdown of Definition df-phl

Step	Hyp	Ref	Expression
1		cphl 21615	. 2 class PreHil
2		vf	. . . . . . . . 9 setvar 𝑓
3	2	cv 1533	. . . . . . . 8 class 𝑓
4		csr 20812	. . . . . . . 8 class *-Ring
5	3, 4	wcel 2099	. . . . . . 7 wff 𝑓 ∈ *-Ring
6		vy	. . . . . . . . . . 11 setvar 𝑦
7		vv	. . . . . . . . . . . 12 setvar 𝑣
8	7	cv 1533	. . . . . . . . . . 11 class 𝑣
9	6	cv 1533	. . . . . . . . . . . 12 class 𝑦
10		vx	. . . . . . . . . . . . 13 setvar 𝑥
11	10	cv 1533	. . . . . . . . . . . 12 class 𝑥
12		vh	. . . . . . . . . . . . 13 setvar ℎ
13	12	cv 1533	. . . . . . . . . . . 12 class ℎ
14	9, 11, 13	co 7415	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
15	6, 8, 14	cmpt 5228	. . . . . . . . . 10 class (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥))
16		vg	. . . . . . . . . . . 12 setvar 𝑔
17	16	cv 1533	. . . . . . . . . . 11 class 𝑔
18		crglmod 21145	. . . . . . . . . . . 12 class ringLMod
19	3, 18	cfv 6545	. . . . . . . . . . 11 class (ringLMod‘𝑓)
20		clmhm 20992	. . . . . . . . . . 11 class LMHom
21	17, 19, 20	co 7415	. . . . . . . . . 10 class (𝑔 LMHom (ringLMod‘𝑓))
22	15, 21	wcel 2099	. . . . . . . . 9 wff (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓))
23	11, 11, 13	co 7415	. . . . . . . . . . 11 class (𝑥ℎ𝑥)
24		c0g 17448	. . . . . . . . . . . 12 class 0g
25	3, 24	cfv 6545	. . . . . . . . . . 11 class (0g‘𝑓)
26	23, 25	wceq 1534	. . . . . . . . . 10 wff (𝑥ℎ𝑥) = (0g‘𝑓)
27	17, 24	cfv 6545	. . . . . . . . . . 11 class (0g‘𝑔)
28	11, 27	wceq 1534	. . . . . . . . . 10 wff 𝑥 = (0g‘𝑔)
29	26, 28	wi 4	. . . . . . . . 9 wff ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔))
30	11, 9, 13	co 7415	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
31		cstv 17262	. . . . . . . . . . . . 13 class *𝑟
32	3, 31	cfv 6545	. . . . . . . . . . . 12 class (*𝑟‘𝑓)
33	30, 32	cfv 6545	. . . . . . . . . . 11 class ((*𝑟‘𝑓)‘(𝑥ℎ𝑦))
34	33, 14	wceq 1534	. . . . . . . . . 10 wff ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
35	34, 6, 8	wral 3051	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
36	22, 29, 35	w3a 1084	. . . . . . . 8 wff ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥))
37	36, 10, 8	wral 3051	. . . . . . 7 wff ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥))
38	5, 37	wa 394	. . . . . 6 wff (𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))
39		csca 17263	. . . . . . 7 class Scalar
40	17, 39	cfv 6545	. . . . . 6 class (Scalar‘𝑔)
41	38, 2, 40	wsbc 3777	. . . . 5 wff [(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))
42		cip 17265	. . . . . 6 class ·𝑖
43	17, 42	cfv 6545	. . . . 5 class (·𝑖‘𝑔)
44	41, 12, 43	wsbc 3777	. . . 4 wff [(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))
45		cbs 17207	. . . . 5 class Base
46	17, 45	cfv 6545	. . . 4 class (Base‘𝑔)
47	44, 7, 46	wsbc 3777	. . 3 wff [(Base‘𝑔) / 𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))
48		clvec 21075	. . 3 class LVec
49	47, 16, 48	crab 3420	. 2 class {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))}
50	1, 49	wceq 1534	1 wff PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))}

This definition is referenced by:  isphl  21619