Theorem isphld 21673

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 2841	. 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ *-Ring)
5		oveq1 7439	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦(·𝑖‘𝑊)𝑥) = (𝑤(·𝑖‘𝑊)𝑥))
6	5	cbvmptv 5254	. . . . 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 1122	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾))
9		isphld.v	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
10	9	eleq2d 2826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑊)))
11	9	eleq2d 2826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑊)))
12	10, 11	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
13		isphld.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊))
14	13	oveqd 7449	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥𝐼𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
15		isphld.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 = (Base‘𝐹))
16	2	fveq2d 6909	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
17	15, 16	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
18	14, 17	eleq12d 2834	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥𝐼𝑦) ∈ 𝐾 ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
19	8, 12, 18	3imtr3d 293	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
20	19	impl 455	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
21	20	an32s 652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
22		oveq1 7439	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
23	22	cbvmptv 5254	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖‘𝑊)𝑦))
24	21, 23	fmptd 7133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
25	24	ralrimiva 3145	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
26		oveq2 7440	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑤(·𝑖‘𝑊)𝑦) = (𝑤(·𝑖‘𝑊)𝑧))
27	26	mpteq2dv 5243	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑦)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)))
28	27	feq1d 6719	. . . . . . . . . 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 3620	. . . . . . . . 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 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
31		eqidd 2737	. . . . . . . 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 1119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))))
34	17	eleq2d 2826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ 𝐾 ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
35		3anrot 1099	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
36	9	eleq2d 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑧 ∈ 𝑉 ↔ 𝑧 ∈ (Base‘𝑊)))
37	36, 10, 11	3anbi123d 1437	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
38	35, 37	bitr3id 285	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑞 · 𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
42		eqidd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑦 = 𝑦)
43	39, 41, 42	oveq123d 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑞 · 𝑥) + 𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
44		eqidd 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑧 = 𝑧)
45	13, 43, 44	oveq123d 7453	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (+g‘𝐹) = (+g‘(Scalar‘𝑊)))
48	46, 47	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
49		isphld.t	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → × = (.r‘𝐹))
50	2	fveq2d 6909	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (.r‘𝐹) = (.r‘(Scalar‘𝑊)))
51	49, 50	eqtrd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
52		eqidd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑞 = 𝑞)
53	13	oveqd 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥𝐼𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
54	51, 52, 53	oveq123d 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑞 × (𝑥𝐼𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
55	13	oveqd 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑦𝐼𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
56	48, 54, 55	oveq123d 7453	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
57	45, 56	eqeq12d 2752	. . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 1354	. . . . . . . . . . . 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 451	. . . . . . . . . . 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 1349	. . . . . . . . . 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 21106	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
65	1, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ LMod)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → 𝑊 ∈ LMod)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
68		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑊) = (Base‘𝑊)
69		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
70	68, 69	lss1 20937	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
73		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
74		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+g‘𝑊) = (+g‘𝑊)
75		eqid 2736	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
76	72, 73, 74, 75, 69	lsscl 20941	. . . . . . . . . . . 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 588	. . . . . . . . . . 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 7439	. . . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧))
80		ovex 7465	. . . . . . . . . . . 12 ⊢ (𝑤(·𝑖‘𝑊)𝑧) ∈ V
81	78, 79, 80	fvmpt3i 7020	. . . . . . . . . . 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 1195	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
84		oveq1 7439	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
85	84, 79, 80	fvmpt3i 7020	. . . . . . . . . . . . 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 7448	. . . . . . . . . . 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 1196	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
89		oveq1 7439	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
90	89, 79, 80	fvmpt3i 7020	. . . . . . . . . . . 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 7450	. . . . . . . . . 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 2786	. . . . . . . . 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 3204	. . . . . . . 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 20867	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
96		rlmlmod 21211	. . . . . . . . . . 11 ⊢ ((Scalar‘𝑊) ∈ Ring → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
97	65, 95, 96	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
98	97	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
99		rlmbas 21201	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(ringLMod‘(Scalar‘𝑊)))
100		fvex 6918	. . . . . . . . . . 11 ⊢ (Scalar‘𝑊) ∈ V
101		rlmsca 21206	. . . . . . . . . . 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 21202	. . . . . . . . . 10 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(ringLMod‘(Scalar‘𝑊)))
104		rlmvsca 21208	. . . . . . . . . 10 ⊢ (.r‘(Scalar‘𝑊)) = ( ·𝑠 ‘(ringLMod‘(Scalar‘𝑊)))
105	68, 99, 72, 102, 73, 74, 103, 75, 104	islmhm2 21038	. . . . . . . . 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 584	. . . . . . . 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 1342	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
108	107	ralrimiva 3145	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
109		oveq2 7440	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑤(·𝑖‘𝑊)𝑧) = (𝑤(·𝑖‘𝑊)𝑥))
110	109	mpteq2dv 5243	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)))
111	110	eleq1d 2825	. . . . . . 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 3620	. . . . . 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 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
114	6, 113	eqeltrid 2844	. . . 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 1119	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → ((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 )))
117	13	oveqd 7449	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐼𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
118		isphld.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 = (0g‘𝐹))
119	2	fveq2d 6909	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝑊)))
120	118, 119	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → 𝑂 = (0g‘(Scalar‘𝑊)))
121	117, 120	eqeq12d 2752	. . . . . . 7 ⊢ (𝜑 → ((𝑥𝐼𝑥) = 𝑂 ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
122		isphld.z	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝑊))
123	122	eqeq2d 2747	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑊)))
124	121, 123	imbi12d 344	. . . . . 6 ⊢ (𝜑 → (((𝑥𝐼𝑥) = 𝑂 → 𝑥 = 0 ) ↔ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
125	116, 10, 124	3imtr3d 293	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑊) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊))))
126	125	imp 406	. . . 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 1122	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)))
129		isphld.c	. . . . . . . . . 10 ⊢ (𝜑 → ∗ = (*𝑟‘𝐹))
130	2	fveq2d 6909	. . . . . . . . . 10 ⊢ (𝜑 → (*𝑟‘𝐹) = (*𝑟‘(Scalar‘𝑊)))
131	129, 130	eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → ∗ = (*𝑟‘(Scalar‘𝑊)))
132	131, 14	fveq12d 6912	. . . . . . . 8 ⊢ (𝜑 → ( ∗ ‘(𝑥𝐼𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
133	13	oveqd 7449	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐼𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
134	132, 133	eqeq12d 2752	. . . . . . 7 ⊢ (𝜑 → (( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥) ↔ ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
135	128, 12, 134	3imtr3d 293	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
136	135	expdimp 452	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
137	136	ralrimiv 3144	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑊)) → ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
138	114, 126, 137	3jca 1128	. . 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 3145	. 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 2736	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
141		eqid 2736	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
142		eqid 2736	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
143		eqid 2736	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
144	68, 72, 140, 141, 142, 143	isphl 21647	. 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 1343	1 ⊢ (𝜑 → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ↦ cmpt 5224  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  ._rcmulr 17299  *_𝑟cstv 17300  Scalarcsca 17301   ·_𝑠 cvsca 17302  ·_𝑖cip 17303  0_gc0g 17485  Ringcrg 20231  *-Ringcsr 20840  LModclmod 20859  LSubSpclss 20930   LMHom clmhm 21019  LVecclvec 21102  ringLModcrglmod 21172  PreHilcphl 21643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-subg 19142  df-ghm 19232  df-mgp 20139  df-ur 20180  df-ring 20233  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lmhm 21022  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-phl 21645

This theorem is referenced by:  phlssphl  21678  frlmphl  21802  hlhilphllem  41966