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Theorem isphld 19931
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 2699 . 2 (𝜑 → (Scalar‘𝑊) ∈ *-Ring)
5 oveq1 6617 . . . . . 6 (𝑦 = 𝑤 → (𝑦(·𝑖𝑊)𝑥) = (𝑤(·𝑖𝑊)𝑥))
65cbvmptv 4715 . . . . 5 (𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥))
7 isphld.cl . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)
873expib 1265 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾))
9 isphld.v . . . . . . . . . . . . . . . 16 (𝜑𝑉 = (Base‘𝑊))
109eleq2d 2684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝑉𝑥 ∈ (Base‘𝑊)))
119eleq2d 2684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦𝑉𝑦 ∈ (Base‘𝑊)))
1210, 11anbi12d 746 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝑉𝑦𝑉) ↔ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
13 isphld.i . . . . . . . . . . . . . . . 16 (𝜑𝐼 = (·𝑖𝑊))
1413oveqd 6627 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐼𝑦) = (𝑥(·𝑖𝑊)𝑦))
15 isphld.k . . . . . . . . . . . . . . . 16 (𝜑𝐾 = (Base‘𝐹))
162fveq2d 6157 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
1715, 16eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
1814, 17eleq12d 2692 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝐼𝑦) ∈ 𝐾 ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
198, 12, 183imtr3d 282 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
2019impl 649 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
2120an32s 845 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (Base‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
22 oveq1 6617 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑤(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322cbvmptv 4715 . . . . . . . . . . 11 (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦))
2421, 23fmptd 6346 . . . . . . . . . 10 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
2524ralrimiva 2961 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
26 oveq2 6618 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑤(·𝑖𝑊)𝑦) = (𝑤(·𝑖𝑊)𝑧))
2726mpteq2dv 4710 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑦)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)))
2827feq1d 5992 . . . . . . . . . 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 3297 . . . . . . . . 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 488 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)):(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
31 eqidd 2622 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (Scalar‘𝑊) = (Scalar‘𝑊))
32 isphld.d . . . . . . . . . . . . . . . 16 ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))
33323exp 1261 . . . . . . . . . . . . . . 15 (𝜑 → (𝑞𝐾 → ((𝑥𝑉𝑦𝑉𝑧𝑉) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))))
3417eleq2d 2684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑞𝐾𝑞 ∈ (Base‘(Scalar‘𝑊))))
35 3anrot 1041 . . . . . . . . . . . . . . . . 17 ((𝑧𝑉𝑥𝑉𝑦𝑉) ↔ (𝑥𝑉𝑦𝑉𝑧𝑉))
369eleq2d 2684 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑧𝑉𝑧 ∈ (Base‘𝑊)))
3736, 10, 113anbi123d 1396 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑧𝑉𝑥𝑉𝑦𝑉) ↔ (𝑧 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))))
3835, 37syl5bbr 274 . . . . . . . . . . . . . . . 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 6627 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑞 · 𝑥) = (𝑞( ·𝑠𝑊)𝑥))
42 eqidd 2622 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑦 = 𝑦)
4339, 41, 42oveq123d 6631 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑞 · 𝑥) + 𝑦) = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
44 eqidd 2622 . . . . . . . . . . . . . . . . . 18 (𝜑𝑧 = 𝑧)
4513, 43, 44oveq123d 6631 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
46 isphld.p . . . . . . . . . . . . . . . . . . 19 (𝜑 = (+g𝐹))
472fveq2d 6157 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (+g𝐹) = (+g‘(Scalar‘𝑊)))
4846, 47eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (𝜑 = (+g‘(Scalar‘𝑊)))
49 isphld.t . . . . . . . . . . . . . . . . . . . 20 (𝜑× = (.r𝐹))
502fveq2d 6157 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (.r𝐹) = (.r‘(Scalar‘𝑊)))
5149, 50eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 (𝜑× = (.r‘(Scalar‘𝑊)))
52 eqidd 2622 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑞 = 𝑞)
5313oveqd 6627 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝐼𝑧) = (𝑥(·𝑖𝑊)𝑧))
5451, 52, 53oveq123d 6631 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑞 × (𝑥𝐼𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
5513oveqd 6627 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑦𝐼𝑧) = (𝑦(·𝑖𝑊)𝑧))
5648, 54, 55oveq123d 6631 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
5745, 56eqeq12d 2636 . . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 1282 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 1279 . . . . . . . . . 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 19038 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . 14 (Base‘𝑊) = (Base‘𝑊)
69 eqid 2621 . . . . . . . . . . . . . 14 (LSubSp‘𝑊) = (LSubSp‘𝑊)
7068, 69lss1 18871 . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 13 (Scalar‘𝑊) = (Scalar‘𝑊)
73 eqid 2621 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
74 eqid 2621 . . . . . . . . . . . . 13 (+g𝑊) = (+g𝑊)
75 eqid 2621 . . . . . . . . . . . . 13 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7672, 73, 74, 75, 69lsscl 18875 . . . . . . . . . . . 12 (((Base‘𝑊) ∈ (LSubSp‘𝑊) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) ∈ (Base‘𝑊))
7771, 76sylancom 700 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) ∈ (Base‘𝑊))
78 oveq1 6617 . . . . . . . . . . . 12 (𝑤 = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦) → (𝑤(·𝑖𝑊)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
79 eqid 2621 . . . . . . . . . . . 12 (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧))
80 ovex 6638 . . . . . . . . . . . 12 (𝑤(·𝑖𝑊)𝑧) ∈ V
8178, 79, 80fvmpt3i 6249 . . . . . . . . . . 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 1066 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
84 oveq1 6617 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤(·𝑖𝑊)𝑧) = (𝑥(·𝑖𝑊)𝑧))
8584, 79, 80fvmpt3i 6249 . . . . . . . . . . . . 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 6626 . . . . . . . . . . 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 1067 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (Base‘𝑊)) ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
89 oveq1 6617 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝑤(·𝑖𝑊)𝑧) = (𝑦(·𝑖𝑊)𝑧))
9089, 79, 80fvmpt3i 6249 . . . . . . . . . . . 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 6628 . . . . . . . . . 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 2665 . . . . . . . . 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 2967 . . . . . . . 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 18803 . . . . . . . . . . 11 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
96 rlmlmod 19137 . . . . . . . . . . 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 19127 . . . . . . . . . 10 (Base‘(Scalar‘𝑊)) = (Base‘(ringLMod‘(Scalar‘𝑊)))
100 fvex 6163 . . . . . . . . . . 11 (Scalar‘𝑊) ∈ V
101 rlmsca 19132 . . . . . . . . . . 11 ((Scalar‘𝑊) ∈ V → (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊))))
102100, 101ax-mp 5 . . . . . . . . . 10 (Scalar‘𝑊) = (Scalar‘(ringLMod‘(Scalar‘𝑊)))
103 rlmplusg 19128 . . . . . . . . . 10 (+g‘(Scalar‘𝑊)) = (+g‘(ringLMod‘(Scalar‘𝑊)))
104 rlmvsca 19134 . . . . . . . . . 10 (.r‘(Scalar‘𝑊)) = ( ·𝑠 ‘(ringLMod‘(Scalar‘𝑊)))
10568, 99, 72, 102, 73, 74, 103, 75, 104islmhm2 18970 . . . . . . . . 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 692 . . . . . . . 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 1243 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
108107ralrimiva 2961 . . . . . 6 (𝜑 → ∀𝑧 ∈ (Base‘𝑊)(𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
109 oveq2 6618 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑤(·𝑖𝑊)𝑧) = (𝑤(·𝑖𝑊)𝑥))
110109mpteq2dv 4710 . . . . . . . 8 (𝑧 = 𝑥 → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑧)) = (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)))
111110eleq1d 2683 . . . . . . 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 3297 . . . . . 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 488 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑊)) → (𝑤 ∈ (Base‘𝑊) ↦ (𝑤(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))))
1146, 113syl5eqel 2702 . . . 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 1261 . . . . . 6 (𝜑 → (𝑥𝑉 → ((𝑥𝐼𝑥) = 𝑂𝑥 = 0 )))
11713oveqd 6627 . . . . . . . 8 (𝜑 → (𝑥𝐼𝑥) = (𝑥(·𝑖𝑊)𝑥))
118 isphld.o . . . . . . . . 9 (𝜑𝑂 = (0g𝐹))
1192fveq2d 6157 . . . . . . . . 9 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝑊)))
120118, 119eqtrd 2655 . . . . . . . 8 (𝜑𝑂 = (0g‘(Scalar‘𝑊)))
121117, 120eqeq12d 2636 . . . . . . 7 (𝜑 → ((𝑥𝐼𝑥) = 𝑂 ↔ (𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
122 isphld.z . . . . . . . 8 (𝜑0 = (0g𝑊))
123122eqeq2d 2631 . . . . . . 7 (𝜑 → (𝑥 = 0𝑥 = (0g𝑊)))
124121, 123imbi12d 334 . . . . . 6 (𝜑 → (((𝑥𝐼𝑥) = 𝑂𝑥 = 0 ) ↔ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊))))
125116, 10, 1243imtr3d 282 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑊) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊))))
126125imp 445 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)))
127 isphld.cj . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))
1281273expib 1265 . . . . . . 7 (𝜑 → ((𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)))
129 isphld.c . . . . . . . . . 10 (𝜑 = (*𝑟𝐹))
1302fveq2d 6157 . . . . . . . . . 10 (𝜑 → (*𝑟𝐹) = (*𝑟‘(Scalar‘𝑊)))
131129, 130eqtrd 2655 . . . . . . . . 9 (𝜑 = (*𝑟‘(Scalar‘𝑊)))
132131, 14fveq12d 6159 . . . . . . . 8 (𝜑 → ( ‘(𝑥𝐼𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
13313oveqd 6627 . . . . . . . 8 (𝜑 → (𝑦𝐼𝑥) = (𝑦(·𝑖𝑊)𝑥))
134132, 133eqeq12d 2636 . . . . . . 7 (𝜑 → (( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥) ↔ ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
135128, 12, 1343imtr3d 282 . . . . . 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 2960 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑊)) → ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
138114, 126, 1373jca 1240 . . 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 2961 . 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 2621 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
141 eqid 2621 . . 3 (0g𝑊) = (0g𝑊)
142 eqid 2621 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
143 eqid 2621 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
14468, 72, 140, 141, 142, 143isphl 19905 . 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 1244 1 (𝜑𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cmpt 4678  wf 5848  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  *𝑟cstv 15875  Scalarcsca 15876   ·𝑠 cvsca 15877  ·𝑖cip 15878  0gc0g 16032  Ringcrg 18479  *-Ringcsr 18776  LModclmod 18795  LSubSpclss 18864   LMHom clmhm 18951  LVecclvec 19034  ringLModcrglmod 19101  PreHilcphl 19901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-subg 17523  df-ghm 17590  df-mgp 18422  df-ur 18434  df-ring 18481  df-subrg 18710  df-lmod 18797  df-lss 18865  df-lmhm 18954  df-lvec 19035  df-sra 19104  df-rgmod 19105  df-phl 19903
This theorem is referenced by:  frlmphl  20052  hlhilphllem  36766
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