Definition df-phl 21179

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 21177	. 2 class PreHil
2		vf	. . . . . . . . 9 setvar 𝑓
3	2	cv 1541	. . . . . . . 8 class 𝑓
4		csr 20452	. . . . . . . 8 class *-Ring
5	3, 4	wcel 2107	. . . . . . 7 wff 𝑓 ∈ *-Ring
6		vy	. . . . . . . . . . 11 setvar 𝑦
7		vv	. . . . . . . . . . . 12 setvar 𝑣
8	7	cv 1541	. . . . . . . . . . 11 class 𝑣
9	6	cv 1541	. . . . . . . . . . . 12 class 𝑦
10		vx	. . . . . . . . . . . . 13 setvar 𝑥
11	10	cv 1541	. . . . . . . . . . . 12 class 𝑥
12		vh	. . . . . . . . . . . . 13 setvar ℎ
13	12	cv 1541	. . . . . . . . . . . 12 class ℎ
14	9, 11, 13	co 7409	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
15	6, 8, 14	cmpt 5232	. . . . . . . . . 10 class (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥))
16		vg	. . . . . . . . . . . 12 setvar 𝑔
17	16	cv 1541	. . . . . . . . . . 11 class 𝑔
18		crglmod 20782	. . . . . . . . . . . 12 class ringLMod
19	3, 18	cfv 6544	. . . . . . . . . . 11 class (ringLMod‘𝑓)
20		clmhm 20630	. . . . . . . . . . 11 class LMHom
21	17, 19, 20	co 7409	. . . . . . . . . 10 class (𝑔 LMHom (ringLMod‘𝑓))
22	15, 21	wcel 2107	. . . . . . . . 9 wff (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓))
23	11, 11, 13	co 7409	. . . . . . . . . . 11 class (𝑥ℎ𝑥)
24		c0g 17385	. . . . . . . . . . . 12 class 0g
25	3, 24	cfv 6544	. . . . . . . . . . 11 class (0g‘𝑓)
26	23, 25	wceq 1542	. . . . . . . . . 10 wff (𝑥ℎ𝑥) = (0g‘𝑓)
27	17, 24	cfv 6544	. . . . . . . . . . 11 class (0g‘𝑔)
28	11, 27	wceq 1542	. . . . . . . . . 10 wff 𝑥 = (0g‘𝑔)
29	26, 28	wi 4	. . . . . . . . 9 wff ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔))
30	11, 9, 13	co 7409	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
31		cstv 17199	. . . . . . . . . . . . 13 class *𝑟
32	3, 31	cfv 6544	. . . . . . . . . . . 12 class (*𝑟‘𝑓)
33	30, 32	cfv 6544	. . . . . . . . . . 11 class ((*𝑟‘𝑓)‘(𝑥ℎ𝑦))
34	33, 14	wceq 1542	. . . . . . . . . 10 wff ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
35	34, 6, 8	wral 3062	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)
36	22, 29, 35	w3a 1088	. . . . . . . 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 3062	. . . . . . 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 397	. . . . . 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 17200	. . . . . . 7 class Scalar
40	17, 39	cfv 6544	. . . . . 6 class (Scalar‘𝑔)
41	38, 2, 40	wsbc 3778	. . . . 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 17202	. . . . . 6 class ·𝑖
43	17, 42	cfv 6544	. . . . 5 class (·𝑖‘𝑔)
44	41, 12, 43	wsbc 3778	. . . 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 17144	. . . . 5 class Base
46	17, 45	cfv 6544	. . . 4 class (Base‘𝑔)
47	44, 7, 46	wsbc 3778	. . 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 20713	. . 3 class LVec
49	47, 16, 48	crab 3433	. 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 1542	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  21181