Theorem isphld 21764

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 2866	. 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ *-Ring)
5		oveq1 7407	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖‘𝑊)𝑥) = (𝑤(·𝑖‘𝑊)𝑥))
6	5	cbvmptv 5209	. . . . 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 1138	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾))
9		isphld.v	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
10	9	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑊)))
11	9	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑊)))
12	10, 11	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
13		isphld.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊))
14	13	oveqd 7417	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥𝐼𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
15		isphld.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
16	2	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
17	15, 16	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
18	14, 17	eleq12d 2859	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥𝐼𝑦) ∈ 𝐾 ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
19	8, 12, 18	3imtr3d 296	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
20	19	impl 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
21	20	an32s 664	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
22		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
23	22	cbvmptv 5209	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖‘𝑊)𝑦))
24	21, 23	fmptd 7099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
25	24	ralrimiva 3157	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
26		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑤(·𝑖‘𝑊)𝑧))
27	26	mpteq2dv 5199	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)))
28	27	feq1d 6677	. . . . . . . . . 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 3583	. . . . . . . . 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 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
31		eqidd 2766	. . . . . . . 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 1135	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))))
34	17	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
35		3anrot 1115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
36	9	eleq2d 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑧 ∈ 𝑉 ↔ 𝑧 ∈ (Base‘𝑊)))
37	36, 10, 11	3anbi123d 1460	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
38	35, 37	bitr3id 288	. . . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑞 · 𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
42		eqidd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑦 = 𝑦)
43	39, 41, 42	oveq123d 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑞 · 𝑥) + 𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
44		eqidd 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑧 = 𝑧)
45	13, 43, 44	oveq123d 7421	. . . . . . . . . . . . . . . . 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 6875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (+g‘𝐹) = (+g‘(Scalar‘𝑊)))
48	46, 47	eqtrd 2800	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
49		isphld.t	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → × = (.r‘𝐹))
50	2	fveq2d 6875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (.r‘𝐹) = (.r‘(Scalar‘𝑊)))
51	49, 50	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
52		eqidd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑞 = 𝑞)
53	13	oveqd 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥𝐼𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
54	51, 52, 53	oveq123d 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑞 × (𝑥𝐼𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
55	13	oveqd 7417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑦𝐼𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
56	48, 54, 55	oveq123d 7421	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
57	45, 56	eqeq12d 2781	. . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . 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 296	. . . . . . . . . . . . . 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 422	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1366	. . . . . . . . . 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 21196	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
65	1, 64	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ LMod)
66	65	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → 𝑊 ∈ LMod)
67	66	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
68		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑊) = (Base‘𝑊)
69		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
70	68, 69	lss1 21028	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) ∈ (LSubSp‘𝑊))
71	67, 70	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (Base‘𝑊) ∈ (LSubSp‘𝑊))
72		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
73		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
74		eqid 2765	. . . . . . . . . . . . 13 ⊢ (+g‘𝑊) = (+g‘𝑊)
75		eqid 2765	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
76	72, 73, 74, 75, 69	lsscl 21032	. . . . . . . . . . . 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 599	. . . . . . . . . . 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 7407	. . . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))
80		ovex 7433	. . . . . . . . . . . 12 ⊢ (𝑤(·𝑖‘𝑊)𝑧) ∈ V
81	78, 79, 80	fvmpt3i 6985	. . . . . . . . . . 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 18	. . . . . . . . . 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 1212	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
84		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
85	84, 79, 80	fvmpt3i 6985	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑥) = (𝑥(·𝑖‘𝑊)𝑧))
86	83, 85	syl 18	. . . . . . . . . . . 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 7416	. . . . . . . . . . 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 1213	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
89		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
90	89, 79, 80	fvmpt3i 6985	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))‘𝑦) = (𝑦(·𝑖‘𝑊)𝑧))
91	88, 90	syl 18	. . . . . . . . . . 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 7418	. . . . . . . . . 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 2810	. . . . . . . . 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 3211	. . . . . . . 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 20958	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
96		rlmlmod 21293	. . . . . . . . . . 11 ⊢ ((Scalar‘𝑊) ∈ Ring → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
97	65, 95, 96	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
98	97	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
99		rlmbas 21283	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(ringLMod‘(Scalar‘𝑊)))
100		fvex 6884	. . . . . . . . . . 11 ⊢ (Scalar‘𝑊) ∈ V
101		rlmsca 21288	. . . . . . . . . . 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 21284	. . . . . . . . . 10 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(ringLMod‘(Scalar‘𝑊)))
104		rlmvsca 21290	. . . . . . . . . 10 ⊢ (.r‘(Scalar‘𝑊)) = ( ·𝑠 ‘(ringLMod‘(Scalar‘𝑊)))
105	68, 99, 72, 102, 73, 74, 103, 75, 104	islmhm2 21128	. . . . . . . . 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 595	. . . . . . . 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 1359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
108	107	ralrimiva 3157	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
109		oveq2 7408	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑤(·𝑖‘𝑊)𝑥))
110	109	mpteq2dv 5199	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)))
111	110	eleq1d 2850	. . . . . . 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 3583	. . . . . 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 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
114	6, 113	eqeltrid 2869	. . . 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 1135	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → ((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 )))
117	13	oveqd 7417	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐼𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
118		isphld.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 = (0g‘𝐹))
119	2	fveq2d 6875	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝑊)))
120	118, 119	eqtrd 2800	. . . . . . . 8 ⊢ (𝜑 → 𝑂 = (0g‘(Scalar‘𝑊)))
121	117, 120	eqeq12d 2781	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝐼𝑥) = 𝑂 ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
122		isphld.z	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝑊))
123	122	eqeq2d 2776	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑊)))
124	121, 123	imbi12d 347	. . . . . 6 ⊢ (𝜑 → (((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 ) ↔ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
125	116, 10, 124	3imtr3d 296	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑊) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
126	125	imp 411	. . . 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 1138	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)))
129		isphld.c	. . . . . . . . . 10 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹))
130	2	fveq2d 6875	. . . . . . . . . 10 ⊢ (𝜑 → (*𝑟‘𝐹) = (*𝑟‘(Scalar‘𝑊)))
131	129, 130	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → ∗ = (*𝑟‘(Scalar‘𝑊)))
132	131, 14	fveq12d 6878	. . . . . . . 8 ⊢ (𝜑 → ( ∗ ‘(𝑥𝐼𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
133	13	oveqd 7417	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐼𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
134	132, 133	eqeq12d 2781	. . . . . . 7 ⊢ (𝜑 → (( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥) ↔ ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
135	128, 12, 134	3imtr3d 296	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
136	135	expdimp 457	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
137	136	ralrimiv 3156	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
138	114, 126, 137	3jca 1144	. . 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 3157	. 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 2765	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
141		eqid 2765	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
142		eqid 2765	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
143		eqid 2765	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
144	68, 72, 140, 141, 142, 143	isphl 21738	. 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 1360	1 ⊢ (𝜑 → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ↦ cmpt 5186  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301  *_𝑟cstv 17302  Scalarcsca 17303   ·_𝑠 cvsca 17304  ·_𝑖cip 17305  0_gc0g 17482  Ringcrg 20306  *-Ringcsr 20910  LModclmod 20950  LSubSpclss 21021   LMHom clmhm 21109  LVecclvec 21192  ringLModcrglmod 21262  PreHilcphl 21734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-subg 19180  df-ghm 19275  df-mgp 20208  df-ur 20255  df-ring 20308  df-subrg 20646  df-lmod 20952  df-lss 21022  df-lmhm 21112  df-lvec 21193  df-sra 21263  df-rgmod 21264  df-phl 21736

This theorem is referenced by:  phlssphl  21769  frlmphl  21891  hlhilphllem  42595