Theorem isphl 21610

Description: The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
isphl.v	⊢ 𝑉 = (Base‘𝑊)
isphl.f	⊢ 𝐹 = (Scalar‘𝑊)
isphl.h	⊢ , = (·𝑖‘𝑊)
isphl.o	⊢ 0 = (0g‘𝑊)
isphl.i	⊢ ∗ = (*𝑟‘𝐹)
isphl.z	⊢ 𝑍 = (0g‘𝐹)

Assertion

Ref	Expression
isphl	⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))

Distinct variable groups:   𝑥,𝑦,𝑉   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   , (𝑥,𝑦)   ∗ (𝑥,𝑦)   0 (𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem isphl

Dummy variables 𝑓 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6849	. . . 4 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) ∈ V)
2		fvexd 6849	. . . . 5 ⊢ ((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) → (·𝑖‘𝑔) ∈ V)
3		fvexd 6849	. . . . . 6 ⊢ (((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) → (Scalar‘𝑔) ∈ V)
4		id 22	. . . . . . . . 9 ⊢ (𝑓 = (Scalar‘𝑔) → 𝑓 = (Scalar‘𝑔))
5		simpll 772	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) → 𝑔 = 𝑊)
6	5	fveq2d 6838	. . . . . . . . . 10 ⊢ (((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) → (Scalar‘𝑔) = (Scalar‘𝑊))
7		isphl.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
8	6, 7	eqtr4di 2793	. . . . . . . . 9 ⊢ (((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) → (Scalar‘𝑔) = 𝐹)
9	4, 8	sylan9eqr 2797	. . . . . . . 8 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑓 = 𝐹)
10	9	eleq1d 2825	. . . . . . 7 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
11		simpllr 781	. . . . . . . . 9 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = (Base‘𝑔))
12		simplll 780	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑔 = 𝑊)
13	12	fveq2d 6838	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = (Base‘𝑊))
14		isphl.v	. . . . . . . . . 10 ⊢ 𝑉 = (Base‘𝑊)
15	13, 14	eqtr4di 2793	. . . . . . . . 9 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = 𝑉)
16	11, 15	eqtrd 2775	. . . . . . . 8 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = 𝑉)
17		simplr 774	. . . . . . . . . . . . 13 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ℎ = (·𝑖‘𝑔))
18	12	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖‘𝑔) = (·𝑖‘𝑊))
19		isphl.h	. . . . . . . . . . . . . 14 ⊢ , = (·𝑖‘𝑊)
20	18, 19	eqtr4di 2793	. . . . . . . . . . . . 13 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖‘𝑔) = , )
21	17, 20	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ℎ = , )
22	21	oveqd 7380	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦ℎ𝑥) = (𝑦 , 𝑥))
23	16, 22	mpteq12dv 5166	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) = (𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)))
24	9	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (ringLMod‘𝑓) = (ringLMod‘𝐹))
25	12, 24	oveq12d 7381	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑔 LMHom (ringLMod‘𝑓)) = (𝑊 LMHom (ringLMod‘𝐹)))
26	23, 25	eleq12d 2834	. . . . . . . . 9 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ↔ (𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹))))
27	21	oveqd 7380	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥ℎ𝑥) = (𝑥 , 𝑥))
28	9	fveq2d 6838	. . . . . . . . . . . 12 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g‘𝑓) = (0g‘𝐹))
29		isphl.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (0g‘𝐹)
30	28, 29	eqtr4di 2793	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g‘𝑓) = 𝑍)
31	27, 30	eqeq12d 2756	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑥ℎ𝑥) = (0g‘𝑓) ↔ (𝑥 , 𝑥) = 𝑍))
32	12	fveq2d 6838	. . . . . . . . . . . 12 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g‘𝑔) = (0g‘𝑊))
33		isphl.o	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑊)
34	32, 33	eqtr4di 2793	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g‘𝑔) = 0 )
35	34	eqeq2d 2751	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥 = (0g‘𝑔) ↔ 𝑥 = 0 ))
36	31, 35	imbi12d 345	. . . . . . . . 9 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ↔ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 )))
37	9	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟‘𝑓) = (*𝑟‘𝐹))
38		isphl.i	. . . . . . . . . . . . 13 ⊢ ∗ = (*𝑟‘𝐹)
39	37, 38	eqtr4di 2793	. . . . . . . . . . . 12 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟‘𝑓) = ∗ )
40	21	oveqd 7380	. . . . . . . . . . . 12 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥ℎ𝑦) = (𝑥 , 𝑦))
41	39, 40	fveq12d 6841	. . . . . . . . . . 11 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = ( ∗ ‘(𝑥 , 𝑦)))
42	41, 22	eqeq12d 2756	. . . . . . . . . 10 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥) ↔ ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
43	16, 42	raleqbidv 3314	. . . . . . . . 9 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥) ↔ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
44	26, 36, 43	3anbi123d 1444	. . . . . . . 8 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)) ↔ ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
45	16, 44	raleqbidv 3314	. . . . . . 7 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)) ↔ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
46	10, 45	anbi12d 638	. . . . . 6 ⊢ ((((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
47	3, 46	sbcied 3773	. . . . 5 ⊢ (((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) ∧ ℎ = (·𝑖‘𝑔)) → ([(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))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
48	2, 47	sbcied 3773	. . . 4 ⊢ ((𝑔 = 𝑊 ∧ 𝑣 = (Base‘𝑔)) → ([(·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))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
49	1, 48	sbcied 3773	. . 3 ⊢ (𝑔 = 𝑊 → ([(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))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
50		df-phl 21608	. . 3 ⊢ 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)))}
51	49, 50	elrab2 3639	. 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
52		3anass 1100	. 2 ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))) ↔ (𝑊 ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
53	51, 52	bitr4i 279	1 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432  [wsbc 3730   ↦ cmpt 5160  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  *_𝑟cstv 17220  Scalarcsca 17221  ·_𝑖cip 17223  0_gc0g 17400  *-Ringcsr 20817   LMHom clmhm 21016  LVecclvec 21099  ringLModcrglmod 21169  PreHilcphl 21606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-iota 6448  df-fv 6500  df-ov 7366  df-phl 21608

This theorem is referenced by:  phllvec  21611  phlsrng  21613  phllmhm  21614  ipcj  21616  ipeq0  21620  isphld  21636  phlpropd  21637