Definition df-phl 21776

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 21774	. 2 class PreHil
2		vf	. . . . . . . . 9 setvar 𝑓
3	2	cv 1569	. . . . . . . 8 class 𝑓
4		csr 20941	. . . . . . . 8 class *-Ring
5	3, 4	wcel 2143	. . . . . . 7 wff 𝑓 ∈ *-Ring
6		vy	. . . . . . . . . . 11 setvar 𝑦
7		vv	. . . . . . . . . . . 12 setvar 𝑣
8	7	cv 1569	. . . . . . . . . . 11 class 𝑣
9	6	cv 1569	. . . . . . . . . . . 12 class 𝑦
10		vx	. . . . . . . . . . . . 13 setvar 𝑥
11	10	cv 1569	. . . . . . . . . . . 12 class 𝑥
12		vh	. . . . . . . . . . . . 13 setvar ℎ
13	12	cv 1569	. . . . . . . . . . . 12 class ℎ
14	9, 11, 13	co 7410	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
15	6, 8, 14	cmpt 5192	. . . . . . . . . 10 class (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥))
16		vg	. . . . . . . . . . . 12 setvar 𝑔
17	16	cv 1569	. . . . . . . . . . 11 class 𝑔
18		crglmod 21293	. . . . . . . . . . . 12 class ringLMod
19	3, 18	cfv 6536	. . . . . . . . . . 11 class (ringLMod‘𝑓)
20		clmhm 21140	. . . . . . . . . . 11 class LMHom
21	17, 19, 20	co 7410	. . . . . . . . . 10 class (𝑔 LMHom (ringLMod‘𝑓))
22	15, 21	wcel 2143	. . . . . . . . 9 wff (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓))
23	11, 11, 13	co 7410	. . . . . . . . . . 11 class (𝑥ℎ𝑥)
24		c0g 17487	. . . . . . . . . . . 12 class 0g
25	3, 24	cfv 6536	. . . . . . . . . . 11 class (0g‘𝑓)
26	23, 25	wceq 1570	. . . . . . . . . 10 wff (𝑥ℎ𝑥) = (0g‘𝑓)
27	17, 24	cfv 6536	. . . . . . . . . . 11 class (0g‘𝑔)
28	11, 27	wceq 1570	. . . . . . . . . 10 wff 𝑥 = (0g‘𝑔)
29	26, 28	wi 4	. . . . . . . . 9 wff ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔))
30	11, 9, 13	co 7410	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
31		cstv 17307	. . . . . . . . . . . . 13 class *𝑟
32	3, 31	cfv 6536	. . . . . . . . . . . 12 class (*𝑟‘𝑓)
33	30, 32	cfv 6536	. . . . . . . . . . 11 class ((*𝑟‘𝑓)‘(𝑥ℎ𝑦))
34	33, 14	wceq 1570	. . . . . . . . . 10 wff ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
35	34, 6, 8	wral 3079	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
36	22, 29, 35	w3a 1103	. . . . . . . 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 3079	. . . . . . 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 400	. . . . . 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 17308	. . . . . . 7 class Scalar
40	17, 39	cfv 6536	. . . . . 6 class (Scalar‘𝑔)
41	38, 2, 40	wsbc 3744	. . . . 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 17310	. . . . . 6 class ·𝑖
43	17, 42	cfv 6536	. . . . 5 class (·𝑖‘𝑔)
44	41, 12, 43	wsbc 3744	. . . 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 17264	. . . . 5 class Base
46	17, 45	cfv 6536	. . . 4 class (Base‘𝑔)
47	44, 7, 46	wsbc 3744	. . 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 21223	. . 3 class LVec
49	47, 16, 48	crab 3416	. 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 1570	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  21778