Theorem isphld 21686

Description: Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
isphld.v	⊢ (𝜑 → 𝑉 = (Base‘𝑊))
isphld.a	⊢ (𝜑 → + = (+g‘𝑊))
isphld.s	⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
isphld.i	⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊))
isphld.z	⊢ (𝜑 → 0 = (0g‘𝑊))
isphld.f	⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
isphld.k	⊢ (𝜑 → 𝐾 = (Base‘𝐹))
isphld.p	⊢ (𝜑 → ⨣ = (+g‘𝐹))
isphld.t	⊢ (𝜑 → × = (.r‘𝐹))
isphld.c	⊢ (𝜑 → ∗ = (*𝑟‘𝐹))
isphld.o	⊢ (𝜑 → 𝑂 = (0g‘𝐹))
isphld.l	⊢ (𝜑 → 𝑊 ∈ LVec)
isphld.r	⊢ (𝜑 → 𝐹 ∈ *-Ring)
isphld.cl	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)
isphld.d	⊢ ((𝜑 ∧ 𝑞 ∈ 𝐾 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))
isphld.ns	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )
isphld.cj	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))

Assertion

Ref	Expression
isphld	⊢ (𝜑 → 𝑊 ∈ PreHil)

Distinct variable groups:   𝑥,𝑞,𝑦,𝑧,𝜑   𝑊,𝑞,𝑥,𝑦,𝑧

Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑞)   ⨣ (𝑥,𝑦,𝑧,𝑞)   · (𝑥,𝑦,𝑧,𝑞)   × (𝑥,𝑦,𝑧,𝑞)   𝐹(𝑥,𝑦,𝑧,𝑞)   𝐼(𝑥,𝑦,𝑧,𝑞)   ∗ (𝑥,𝑦,𝑧,𝑞)   𝐾(𝑥,𝑦,𝑧,𝑞)   𝑂(𝑥,𝑦,𝑧,𝑞)   𝑉(𝑥,𝑦,𝑧,𝑞)   0 (𝑥,𝑦,𝑧,𝑞)

Proof of Theorem isphld

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isphld.l	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
2		isphld.f	. . 3 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
3		isphld.r	. . 3 ⊢ (𝜑 → 𝐹 ∈ *-Ring)
4	2, 3	eqeltrrd 2862	. 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ *-Ring)
5		oveq1 7399	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖‘𝑊)𝑥) = (𝑤(·𝑖‘𝑊)𝑥))
6	5	cbvmptv 5203	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥))
7		isphld.cl	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)
8	7	3expib 1134	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾))
9		isphld.v	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
10	9	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑊)))
11	9	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑊)))
12	10, 11	anbi12d 641	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
13		isphld.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊))
14	13	oveqd 7409	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥𝐼𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
15		isphld.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
16	2	fveq2d 6867	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
17	15, 16	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
18	14, 17	eleq12d 2855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥𝐼𝑦) ∈ 𝐾 ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
19	8, 12, 18	3imtr3d 295	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
20	19	impl 459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
21	20	an32s 662	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
22		oveq1 7399	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
23	22	cbvmptv 5203	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖‘𝑊)𝑦))
24	21, 23	fmptd 7091	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
25	24	ralrimiva 3153	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
26		oveq2 7400	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑤(·𝑖‘𝑊)𝑧))
27	26	mpteq2dv 5193	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)))
28	27	feq1d 6669	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ↔ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
29	28	rspccva 3580	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
30	25, 29	sylan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
31		eqidd 2762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (Scalar‘𝑊) = (Scalar‘𝑊))
32		isphld.d	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐾 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))
33	32	3exp 1131	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))))
34	17	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
35		3anrot 1111	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
36	9	eleq2d 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑧 ∈ 𝑉 ↔ 𝑧 ∈ (Base‘𝑊)))
37	36, 10, 11	3anbi123d 1456	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
38	35, 37	bitr3id 287	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
39		isphld.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → + = (+g‘𝑊))
40		isphld.s	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
41	40	oveqd 7409	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑞 · 𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
42		eqidd 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑦 = 𝑦)
43	39, 41, 42	oveq123d 7413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑞 · 𝑥) + 𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
44		eqidd 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑧 = 𝑧)
45	13, 43, 44	oveq123d 7413	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧))
46		isphld.p	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⨣ = (+g‘𝐹))
47	2	fveq2d 6867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (+g‘𝐹) = (+g‘(Scalar‘𝑊)))
48	46, 47	eqtrd 2796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
49		isphld.t	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → × = (.r‘𝐹))
50	2	fveq2d 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (.r‘𝐹) = (.r‘(Scalar‘𝑊)))
51	49, 50	eqtrd 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
52		eqidd 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑞 = 𝑞)
53	13	oveqd 7409	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥𝐼𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
54	51, 52, 53	oveq123d 7413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑞 × (𝑥𝐼𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
55	13	oveqd 7409	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑦𝐼𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
56	48, 54, 55	oveq123d 7413	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
57	45, 56	eqeq12d 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)) ↔ (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))
58	38, 57	imbi12d 346	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧))) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))))
59	33, 34, 58	3imtr3d 295	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑞 ∈ (Base‘(Scalar‘𝑊)) → ((𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))))
60	59	imp31 421	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
61	60	3exp2 1367	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊))) → (𝑧 ∈ (Base‘𝑊) → (𝑥 ∈ (Base‘𝑊) → (𝑦 ∈ (Base‘𝑊) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))))
62	61	impancom 455	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑞 ∈ (Base‘(Scalar‘𝑊)) → (𝑥 ∈ (Base‘𝑊) → (𝑦 ∈ (Base‘𝑊) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))))
63	62	3imp2 1362	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
64		lveclmod 21153	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
65	1, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ LMod)
66	65	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → 𝑊 ∈ LMod)
67	66	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
68		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑊) = (Base‘𝑊)
69		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
70	68, 69	lss1 20985	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) ∈ (LSubSp‘𝑊))
71	67, 70	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (Base‘𝑊) ∈ (LSubSp‘𝑊))
72		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
73		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
74		eqid 2761	. . . . . . . . . . . . 13 ⊢ (+g‘𝑊) = (+g‘𝑊)
75		eqid 2761	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
76	72, 73, 74, 75, 69	lsscl 20989	. . . . . . . . . . . 12 ⊢ (((Base‘𝑊) ∈ (LSubSp‘𝑊) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
77	71, 76	sylancom 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
78		oveq1 7399	. . . . . . . . . . . 12 ⊢ (𝑤 = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) → (𝑤(·𝑖‘𝑊)𝑧) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧))
79		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))
80		ovex 7425	. . . . . . . . . . . 12 ⊢ (𝑤(·𝑖‘𝑊)𝑧) ∈ V
81	78, 79, 80	fvmpt3i 6977	. . . . . . . . . . 11 ⊢ (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧))
82	77, 81	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧))
83		simpr2 1208	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
84		oveq1 7399	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
85	84, 79, 80	fvmpt3i 6977	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥) = (𝑥(·𝑖‘𝑊)𝑧))
86	83, 85	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥) = (𝑥(·𝑖‘𝑊)𝑧))
87	86	oveq2d 7408	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
88		simpr3 1209	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
89		oveq1 7399	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
90	89, 79, 80	fvmpt3i 6977	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦) = (𝑦(·𝑖‘𝑊)𝑧))
91	88, 90	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦) = (𝑦(·𝑖‘𝑊)𝑧))
92	87, 91	oveq12d 7410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
93	63, 82, 92	3eqtr4d 2806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦)))
94	93	ralrimivvva 3207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦)))
95	72	lmodring 20915	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
96		rlmlmod 21250	. . . . . . . . . . 11 ⊢ ((Scalar‘𝑊) ∈ Ring → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
97	65, 95, 96	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
98	97	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
99		rlmbas 21240	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(ringLMod‘(Scalar‘𝑊)))
100		fvex 6876	. . . . . . . . . . 11 ⊢ (Scalar‘𝑊) ∈ V
101		rlmsca 21245	. . . . . . . . . . 11 ⊢ ((Scalar‘𝑊) ∈ V → (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊))))
102	100, 101	ax-mp 5	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊)))
103		rlmplusg 21241	. . . . . . . . . 10 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(ringLMod‘(Scalar‘𝑊)))
104		rlmvsca 21247	. . . . . . . . . 10 ⊢ (.r‘(Scalar‘𝑊)) = ( ·𝑠 ‘(ringLMod‘(Scalar‘𝑊)))
105	68, 99, 72, 102, 73, 74, 103, 75, 104	islmhm2 21085	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (ringLMod‘(Scalar‘𝑊)) ∈ LMod) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦)))))
106	66, 98, 105	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦)))))
107	30, 31, 94, 106	mpbir3and 1355	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
108	107	ralrimiva 3153	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
109		oveq2 7400	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑤(·𝑖‘𝑊)𝑥))
110	109	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)))
111	110	eleq1d 2846	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊)))))
112	111	rspccva 3580	. . . . . 6 ⊢ ((∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
113	108, 112	sylan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
114	6, 113	eqeltrid 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
115		isphld.ns	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )
116	115	3exp 1131	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → ((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 )))
117	13	oveqd 7409	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐼𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
118		isphld.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 = (0g‘𝐹))
119	2	fveq2d 6867	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝑊)))
120	118, 119	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → 𝑂 = (0g‘(Scalar‘𝑊)))
121	117, 120	eqeq12d 2777	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝐼𝑥) = 𝑂 ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
122		isphld.z	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝑊))
123	122	eqeq2d 2772	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑊)))
124	121, 123	imbi12d 346	. . . . . 6 ⊢ (𝜑 → (((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 ) ↔ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
125	116, 10, 124	3imtr3d 295	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑊) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
126	125	imp 410	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)))
127		isphld.cj	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))
128	127	3expib 1134	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)))
129		isphld.c	. . . . . . . . . 10 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹))
130	2	fveq2d 6867	. . . . . . . . . 10 ⊢ (𝜑 → (*𝑟‘𝐹) = (*𝑟‘(Scalar‘𝑊)))
131	129, 130	eqtrd 2796	. . . . . . . . 9 ⊢ (𝜑 → ∗ = (*𝑟‘(Scalar‘𝑊)))
132	131, 14	fveq12d 6870	. . . . . . . 8 ⊢ (𝜑 → ( ∗ ‘(𝑥𝐼𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
133	13	oveqd 7409	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐼𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
134	132, 133	eqeq12d 2777	. . . . . . 7 ⊢ (𝜑 → (( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥) ↔ ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
135	128, 12, 134	3imtr3d 295	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
136	135	expdimp 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
137	136	ralrimiv 3152	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
138	114, 126, 137	3jca 1140	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
139	138	ralrimiva 3153	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
140		eqid 2761	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
141		eqid 2761	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
142		eqid 2761	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
143		eqid 2761	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
144	68, 72, 140, 141, 142, 143	isphl 21660	. 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (Scalar‘𝑊) ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))))
145	1, 4, 139, 144	syl3anbrc 1356	1 ⊢ (𝜑 → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ↦ cmpt 5180  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  ._rcmulr 17270  *_𝑟cstv 17271  Scalarcsca 17272   ·_𝑠 cvsca 17273  ·_𝑖cip 17274  0_gc0g 17451  Ringcrg 20262  *-Ringcsr 20867  LModclmod 20907  LSubSpclss 20978   LMHom clmhm 21066  LVecclvec 21149  ringLModcrglmod 21219  PreHilcphl 21656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-subg 19148  df-ghm 19237  df-mgp 20170  df-ur 20211  df-ring 20264  df-subrg 20599  df-lmod 20909  df-lss 20979  df-lmhm 21069  df-lvec 21150  df-sra 21220  df-rgmod 21221  df-phl 21658

This theorem is referenced by:  phlssphl  21691  frlmphl  21813  hlhilphllem  42547