Definition df-phl 21644

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 21642	. 2 class PreHil
2		vf	. . . . . . . . 9 setvar 𝑓
3	2	cv 1539	. . . . . . . 8 class 𝑓
4		csr 20839	. . . . . . . 8 class *-Ring
5	3, 4	wcel 2108	. . . . . . 7 wff 𝑓 ∈ *-Ring
6		vy	. . . . . . . . . . 11 setvar 𝑦
7		vv	. . . . . . . . . . . 12 setvar 𝑣
8	7	cv 1539	. . . . . . . . . . 11 class 𝑣
9	6	cv 1539	. . . . . . . . . . . 12 class 𝑦
10		vx	. . . . . . . . . . . . 13 setvar 𝑥
11	10	cv 1539	. . . . . . . . . . . 12 class 𝑥
12		vh	. . . . . . . . . . . . 13 setvar ℎ
13	12	cv 1539	. . . . . . . . . . . 12 class ℎ
14	9, 11, 13	co 7431	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
15	6, 8, 14	cmpt 5225	. . . . . . . . . 10 class (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥))
16		vg	. . . . . . . . . . . 12 setvar 𝑔
17	16	cv 1539	. . . . . . . . . . 11 class 𝑔
18		crglmod 21171	. . . . . . . . . . . 12 class ringLMod
19	3, 18	cfv 6561	. . . . . . . . . . 11 class (ringLMod‘𝑓)
20		clmhm 21018	. . . . . . . . . . 11 class LMHom
21	17, 19, 20	co 7431	. . . . . . . . . 10 class (𝑔 LMHom (ringLMod‘𝑓))
22	15, 21	wcel 2108	. . . . . . . . 9 wff (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓))
23	11, 11, 13	co 7431	. . . . . . . . . . 11 class (𝑥ℎ𝑥)
24		c0g 17484	. . . . . . . . . . . 12 class 0g
25	3, 24	cfv 6561	. . . . . . . . . . 11 class (0g‘𝑓)
26	23, 25	wceq 1540	. . . . . . . . . 10 wff (𝑥ℎ𝑥) = (0g‘𝑓)
27	17, 24	cfv 6561	. . . . . . . . . . 11 class (0g‘𝑔)
28	11, 27	wceq 1540	. . . . . . . . . 10 wff 𝑥 = (0g‘𝑔)
29	26, 28	wi 4	. . . . . . . . 9 wff ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔))
30	11, 9, 13	co 7431	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
31		cstv 17299	. . . . . . . . . . . . 13 class *𝑟
32	3, 31	cfv 6561	. . . . . . . . . . . 12 class (*𝑟‘𝑓)
33	30, 32	cfv 6561	. . . . . . . . . . 11 class ((*𝑟‘𝑓)‘(𝑥ℎ𝑦))
34	33, 14	wceq 1540	. . . . . . . . . 10 wff ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
35	34, 6, 8	wral 3061	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
36	22, 29, 35	w3a 1087	. . . . . . . 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 3061	. . . . . . 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 395	. . . . . 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 17300	. . . . . . 7 class Scalar
40	17, 39	cfv 6561	. . . . . 6 class (Scalar‘𝑔)
41	38, 2, 40	wsbc 3788	. . . . 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 17302	. . . . . 6 class ·𝑖
43	17, 42	cfv 6561	. . . . 5 class (·𝑖‘𝑔)
44	41, 12, 43	wsbc 3788	. . . 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 17247	. . . . 5 class Base
46	17, 45	cfv 6561	. . . 4 class (Base‘𝑔)
47	44, 7, 46	wsbc 3788	. . 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 21101	. . 3 class LVec
49	47, 16, 48	crab 3436	. 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 1540	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  21646