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Theorem isphld 21043
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 (𝜑𝑉 = (Base‘𝑊))
isphld.a (𝜑+ = (+g𝑊))
isphld.s (𝜑· = ( ·𝑠𝑊))
isphld.i (𝜑𝐼 = (·𝑖𝑊))
isphld.z (𝜑0 = (0g𝑊))
isphld.f (𝜑𝐹 = (Scalar‘𝑊))
isphld.k (𝜑𝐾 = (Base‘𝐹))
isphld.p (𝜑 = (+g𝐹))
isphld.t (𝜑× = (.r𝐹))
isphld.c (𝜑 = (*𝑟𝐹))
isphld.o (𝜑𝑂 = (0g𝐹))
isphld.l (𝜑𝑊 ∈ LVec)
isphld.r (𝜑𝐹 ∈ *-Ring)
isphld.cl ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)
isphld.d ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))
isphld.ns ((𝜑𝑥𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )
isphld.cj ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))
Assertion
Ref Expression
isphld (𝜑𝑊 ∈ 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.
StepHypRef Expression
1 isphld.l . 2 (𝜑𝑊 ∈ LVec)
2 isphld.f . . 3 (𝜑𝐹 = (Scalar‘𝑊))
3 isphld.r . . 3 (𝜑𝐹 ∈ *-Ring)
42, 3eqeltrrd 2839 . 2 (𝜑 → (Scalar‘𝑊) ∈ *-Ring)
5 oveq1 7360 . . . . . 6 (𝑦 = 𝑤 → (𝑦(·𝑖𝑊)𝑥) = (𝑤(·𝑖𝑊)𝑥))
65cbvmptv 5216 . . . . 5 (𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥))
7 isphld.cl . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)
873expib 1122 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾))
9 isphld.v . . . . . . . . . . . . . . . 16 (𝜑𝑉 = (Base‘𝑊))
109eleq2d 2823 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝑉𝑥 ∈ (Base‘𝑊)))
119eleq2d 2823 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝑉𝑦 ∈ (Base‘𝑊)))
1210, 11anbi12d 631 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝑉𝑦𝑉) ↔ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
13 isphld.i . . . . . . . . . . . . . . . 16 (𝜑𝐼 = (·𝑖𝑊))
1413oveqd 7370 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐼𝑦) = (𝑥(·𝑖𝑊)𝑦))
15 isphld.k . . . . . . . . . . . . . . . 16 (𝜑𝐾 = (Base‘𝐹))
162fveq2d 6843 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
1715, 16eqtrd 2776 . . . . . . . . . . . . . . 15 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
1814, 17eleq12d 2832 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝐼𝑦) ∈ 𝐾 ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
198, 12, 183imtr3d 292 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
2019impl 456 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
2120an32s 650 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
22 oveq1 7360 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑤(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322cbvmptv 5216 . . . . . . . . . . 11 (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦))
2421, 23fmptd 7058 . . . . . . . . . 10 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
2524ralrimiva 3141 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
26 oveq2 7361 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑤(·𝑖𝑊)𝑦) = (𝑤(·𝑖𝑊)𝑧))
2726mpteq2dv 5205 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)))
2827feq1d 6650 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ↔ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
2928rspccva 3578 . . . . . . . . 9 ((∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
3025, 29sylan 580 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
31 eqidd 2737 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (Scalar‘𝑊) = (Scalar‘𝑊))
32 isphld.d . . . . . . . . . . . . . . . 16 ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))
33323exp 1119 . . . . . . . . . . . . . . 15 (𝜑 → (𝑞𝐾 → ((𝑥𝑉𝑦𝑉𝑧𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))))
3417eleq2d 2823 . . . . . . . . . . . . . . 15 (𝜑 → (𝑞𝐾𝑞 ∈ (Base‘(Scalar‘𝑊))))
35 3anrot 1100 . . . . . . . . . . . . . . . . 17 ((𝑧𝑉𝑥𝑉𝑦𝑉) ↔ (𝑥𝑉𝑦𝑉𝑧𝑉))
369eleq2d 2823 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑧𝑉𝑧 ∈ (Base‘𝑊)))
3736, 10, 113anbi123d 1436 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑧𝑉𝑥𝑉𝑦𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
3835, 37bitr3id 284 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑥𝑉𝑦𝑉𝑧𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
39 isphld.a . . . . . . . . . . . . . . . . . . 19 (𝜑+ = (+g𝑊))
40 isphld.s . . . . . . . . . . . . . . . . . . . 20 (𝜑· = ( ·𝑠𝑊))
4140oveqd 7370 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑞 · 𝑥) = (𝑞( ·𝑠𝑊)𝑥))
42 eqidd 2737 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑦 = 𝑦)
4339, 41, 42oveq123d 7374 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑞 · 𝑥) + 𝑦) = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
44 eqidd 2737 . . . . . . . . . . . . . . . . . 18 (𝜑𝑧 = 𝑧)
4513, 43, 44oveq123d 7374 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
46 isphld.p . . . . . . . . . . . . . . . . . . 19 (𝜑 = (+g𝐹))
472fveq2d 6843 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (+g𝐹) = (+g‘(Scalar‘𝑊)))
4846, 47eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑 = (+g‘(Scalar‘𝑊)))
49 isphld.t . . . . . . . . . . . . . . . . . . . 20 (𝜑× = (.r𝐹))
502fveq2d 6843 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (.r𝐹) = (.r‘(Scalar‘𝑊)))
5149, 50eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 (𝜑× = (.r‘(Scalar‘𝑊)))
52 eqidd 2737 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑞 = 𝑞)
5313oveqd 7370 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝐼𝑧) = (𝑥(·𝑖𝑊)𝑧))
5451, 52, 53oveq123d 7374 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑞 × (𝑥𝐼𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
5513oveqd 7370 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑦𝐼𝑧) = (𝑦(·𝑖𝑊)𝑧))
5648, 54, 55oveq123d 7374 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
5745, 56eqeq12d 2752 . . . . . . . . . . . . . . . 16 (𝜑 → ((((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)) ↔ (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))
5838, 57imbi12d 344 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑥𝑉𝑦𝑉𝑧𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧))) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))))
5933, 34, 583imtr3d 292 . . . . . . . . . . . . . 14 (𝜑 → (𝑞 ∈ (Base‘(Scalar‘𝑊)) → ((𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))))
6059imp31 418 . . . . . . . . . . . . 13 (((𝜑𝑞 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
61603exp2 1354 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ (Base‘(Scalar‘𝑊))) → (𝑧 ∈ (Base‘𝑊) → (𝑥 ∈ (Base‘𝑊) → (𝑦 ∈ (Base‘𝑊) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))))
6261impancom 452 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝑞 ∈ (Base‘(Scalar‘𝑊)) → (𝑥 ∈ (Base‘𝑊) → (𝑦 ∈ (Base‘𝑊) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))))
63623imp2 1349 . . . . . . . . . 10 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
64 lveclmod 20552 . . . . . . . . . . . . . . . 16 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
651, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑊 ∈ LMod)
6665adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ (Base‘𝑊)) → 𝑊 ∈ LMod)
6766adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ (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)
7068, 69lss1 20384 . . . . . . . . . . . . 13 (𝑊 ∈ LMod → (Base‘𝑊) ∈ (LSubSp‘𝑊))
7167, 70syl 17 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (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 (+g𝑊) = (+g𝑊)
75 eqid 2736 . . . . . . . . . . . . 13 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7672, 73, 74, 75, 69lsscl 20388 . . . . . . . . . . . 12 (((Base‘𝑊) ∈ (LSubSp‘𝑊) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) ∈ (Base‘𝑊))
7771, 76sylancom 588 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) ∈ (Base‘𝑊))
78 oveq1 7360 . . . . . . . . . . . 12 (𝑤 = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) → (𝑤(·𝑖𝑊)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
79 eqid 2736 . . . . . . . . . . . 12 (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))
80 ovex 7386 . . . . . . . . . . . 12 (𝑤(·𝑖𝑊)𝑧) ∈ V
8178, 79, 80fvmpt3i 6950 . . . . . . . . . . 11 (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
8277, 81syl 17 . . . . . . . . . 10 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
83 simpr2 1195 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
84 oveq1 7360 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤(·𝑖𝑊)𝑧) = (𝑥(·𝑖𝑊)𝑧))
8584, 79, 80fvmpt3i 6950 . . . . . . . . . . . . 13 (𝑥 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥) = (𝑥(·𝑖𝑊)𝑧))
8683, 85syl 17 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥) = (𝑥(·𝑖𝑊)𝑧))
8786oveq2d 7369 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
88 simpr3 1196 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
89 oveq1 7360 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝑤(·𝑖𝑊)𝑧) = (𝑦(·𝑖𝑊)𝑧))
9089, 79, 80fvmpt3i 6950 . . . . . . . . . . . 12 (𝑦 ∈ (Base‘𝑊) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦) = (𝑦(·𝑖𝑊)𝑧))
9188, 90syl 17 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦) = (𝑦(·𝑖𝑊)𝑧))
9287, 91oveq12d 7371 . . . . . . . . . 10 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
9363, 82, 923eqtr4d 2786 . . . . . . . . 9 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦)))
9493ralrimivvva 3198 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦)))
9572lmodring 20315 . . . . . . . . . . 11 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
96 rlmlmod 20659 . . . . . . . . . . 11 ((Scalar‘𝑊) ∈ Ring → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
9765, 95, 963syl 18 . . . . . . . . . 10 (𝜑 → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
9897adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (Base‘𝑊)) → (ringLMod‘(Scalar‘𝑊)) ∈ LMod)
99 rlmbas 20649 . . . . . . . . . 10 (Base‘(Scalar‘𝑊)) = (Base‘(ringLMod‘(Scalar‘𝑊)))
100 fvex 6852 . . . . . . . . . . 11 (Scalar‘𝑊) ∈ V
101 rlmsca 20654 . . . . . . . . . . 11 ((Scalar‘𝑊) ∈ V → (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊))))
102100, 101ax-mp 5 . . . . . . . . . 10 (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊)))
103 rlmplusg 20650 . . . . . . . . . 10 (+g‘(Scalar‘𝑊)) = (+g‘(ringLMod‘(Scalar‘𝑊)))
104 rlmvsca 20656 . . . . . . . . . 10 (.r‘(Scalar‘𝑊)) = ( ·𝑠 ‘(ringLMod‘(Scalar‘𝑊)))
10568, 99, 72, 102, 73, 74, 103, 75, 104islmhm2 20484 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (ringLMod‘(Scalar‘𝑊)) ∈ LMod) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦)))))
10666, 98, 105syl2anc 584 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑞(.r‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑥))(+g‘(Scalar‘𝑊))((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))‘𝑦)))))
10730, 31, 94, 106mpbir3and 1342 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
108107ralrimiva 3141 . . . . . 6 (𝜑 → ∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
109 oveq2 7361 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑤(·𝑖𝑊)𝑧) = (𝑤(·𝑖𝑊)𝑥))
110109mpteq2dv 5205 . . . . . . . 8 (𝑧 = 𝑥 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)))
111110eleq1d 2822 . . . . . . 7 (𝑧 = 𝑥 → ((𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ↔ (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊)))))
112111rspccva 3578 . . . . . 6 ((∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
113108, 112sylan 580 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
1146, 113eqeltrid 2842 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
115 isphld.ns . . . . . . 7 ((𝜑𝑥𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )
1161153exp 1119 . . . . . 6 (𝜑 → (𝑥𝑉 → ((𝑥𝐼𝑥) = 𝑂𝑥 = 0 )))
11713oveqd 7370 . . . . . . . 8 (𝜑 → (𝑥𝐼𝑥) = (𝑥(·𝑖𝑊)𝑥))
118 isphld.o . . . . . . . . 9 (𝜑𝑂 = (0g𝐹))
1192fveq2d 6843 . . . . . . . . 9 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝑊)))
120118, 119eqtrd 2776 . . . . . . . 8 (𝜑𝑂 = (0g‘(Scalar‘𝑊)))
121117, 120eqeq12d 2752 . . . . . . 7 (𝜑 → ((𝑥𝐼𝑥) = 𝑂 ↔ (𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
122 isphld.z . . . . . . . 8 (𝜑0 = (0g𝑊))
123122eqeq2d 2747 . . . . . . 7 (𝜑 → (𝑥 = 0𝑥 = (0g𝑊)))
124121, 123imbi12d 344 . . . . . 6 (𝜑 → (((𝑥𝐼𝑥) = 𝑂𝑥 = 0 ) ↔ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊))))
125116, 10, 1243imtr3d 292 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑊) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊))))
126125imp 407 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)))
127 isphld.cj . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))
1281273expib 1122 . . . . . . 7 (𝜑 → ((𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)))
129 isphld.c . . . . . . . . . 10 (𝜑 = (*𝑟𝐹))
1302fveq2d 6843 . . . . . . . . . 10 (𝜑 → (*𝑟𝐹) = (*𝑟‘(Scalar‘𝑊)))
131129, 130eqtrd 2776 . . . . . . . . 9 (𝜑 = (*𝑟‘(Scalar‘𝑊)))
132131, 14fveq12d 6846 . . . . . . . 8 (𝜑 → ( ‘(𝑥𝐼𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
13313oveqd 7370 . . . . . . . 8 (𝜑 → (𝑦𝐼𝑥) = (𝑦(·𝑖𝑊)𝑥))
134132, 133eqeq12d 2752 . . . . . . 7 (𝜑 → (( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥) ↔ ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
135128, 12, 1343imtr3d 292 . . . . . 6 (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
136135expdimp 453 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑊)) → (𝑦 ∈ (Base‘𝑊) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
137136ralrimiv 3140 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑊)) → ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
138114, 126, 1373jca 1128 . . 3 ((𝜑𝑥 ∈ (Base‘𝑊)) → ((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
139138ralrimiva 3141 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
140 eqid 2736 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
141 eqid 2736 . . 3 (0g𝑊) = (0g𝑊)
142 eqid 2736 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
143 eqid 2736 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
14468, 72, 140, 141, 142, 143isphl 21017 . 2 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (Scalar‘𝑊) ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))))
1451, 4, 139, 144syl3anbrc 1343 1 (𝜑𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  Vcvv 3443  cmpt 5186  wf 6489  cfv 6493  (class class class)co 7353  Basecbs 17075  +gcplusg 17125  .rcmulr 17126  *𝑟cstv 17127  Scalarcsca 17128   ·𝑠 cvsca 17129  ·𝑖cip 17130  0gc0g 17313  Ringcrg 19950  *-Ringcsr 20288  LModclmod 20307  LSubSpclss 20377   LMHom clmhm 20465  LVecclvec 20548  ringLModcrglmod 20615  PreHilcphl 21013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-ress 17105  df-plusg 17138  df-mulr 17139  df-sca 17141  df-vsca 17142  df-ip 17143  df-0g 17315  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-subg 18916  df-ghm 18997  df-mgp 19888  df-ur 19905  df-ring 19952  df-subrg 20205  df-lmod 20309  df-lss 20378  df-lmhm 20468  df-lvec 20549  df-sra 20618  df-rgmod 20619  df-phl 21015
This theorem is referenced by:  phlssphl  21048  frlmphl  21172  hlhilphllem  40393
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