Definition df-phl 21586

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 21584	. 2 class PreHil
2		vf	. . . . . . . . 9 setvar 𝑓
3	2	cv 1539	. . . . . . . 8 class 𝑓
4		csr 20798	. . . . . . . 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 7405	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
15	6, 8, 14	cmpt 5201	. . . . . . . . . 10 class (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥))
16		vg	. . . . . . . . . . . 12 setvar 𝑔
17	16	cv 1539	. . . . . . . . . . 11 class 𝑔
18		crglmod 21130	. . . . . . . . . . . 12 class ringLMod
19	3, 18	cfv 6531	. . . . . . . . . . 11 class (ringLMod‘𝑓)
20		clmhm 20977	. . . . . . . . . . 11 class LMHom
21	17, 19, 20	co 7405	. . . . . . . . . 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 7405	. . . . . . . . . . 11 class (𝑥ℎ𝑥)
24		c0g 17453	. . . . . . . . . . . 12 class 0g
25	3, 24	cfv 6531	. . . . . . . . . . 11 class (0g‘𝑓)
26	23, 25	wceq 1540	. . . . . . . . . 10 wff (𝑥ℎ𝑥) = (0g‘𝑓)
27	17, 24	cfv 6531	. . . . . . . . . . 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 7405	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
31		cstv 17273	. . . . . . . . . . . . 13 class *𝑟
32	3, 31	cfv 6531	. . . . . . . . . . . 12 class (*𝑟‘𝑓)
33	30, 32	cfv 6531	. . . . . . . . . . 11 class ((*𝑟‘𝑓)‘(𝑥ℎ𝑦))
34	33, 14	wceq 1540	. . . . . . . . . 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 1086	. . . . . . . 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 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 17274	. . . . . . 7 class Scalar
40	17, 39	cfv 6531	. . . . . 6 class (Scalar‘𝑔)
41	38, 2, 40	wsbc 3765	. . . . 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 17276	. . . . . 6 class ·𝑖
43	17, 42	cfv 6531	. . . . 5 class (·𝑖‘𝑔)
44	41, 12, 43	wsbc 3765	. . . 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 17228	. . . . 5 class Base
46	17, 45	cfv 6531	. . . 4 class (Base‘𝑔)
47	44, 7, 46	wsbc 3765	. . 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 21060	. . 3 class LVec
49	47, 16, 48	crab 3415	. 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  21588