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Theorem isphl 19792
 Description: The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
isphl.v 𝑉 = (Base‘𝑊)
isphl.f 𝐹 = (Scalar‘𝑊)
isphl.h , = (·𝑖𝑊)
isphl.o 0 = (0g𝑊)
isphl.i = (*𝑟𝐹)
isphl.z 𝑍 = (0g𝐹)
Assertion
Ref Expression
isphl (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
Distinct variable groups:   𝑥,𝑦,𝑉   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   , (𝑥,𝑦)   (𝑥,𝑦)   0 (𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem isphl
Dummy variables 𝑓 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6113 . . . . 5 (Base‘𝑔) ∈ V
21a1i 11 . . . 4 (𝑔 = 𝑊 → (Base‘𝑔) ∈ V)
3 fvex 6113 . . . . . 6 (·𝑖𝑔) ∈ V
43a1i 11 . . . . 5 ((𝑔 = 𝑊𝑣 = (Base‘𝑔)) → (·𝑖𝑔) ∈ V)
5 fvex 6113 . . . . . . 7 (Scalar‘𝑔) ∈ V
65a1i 11 . . . . . 6 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) ∈ V)
7 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑔) → 𝑓 = (Scalar‘𝑔))
8 simpll 786 . . . . . . . . . . 11 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → 𝑔 = 𝑊)
98fveq2d 6107 . . . . . . . . . 10 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = (Scalar‘𝑊))
10 isphl.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
119, 10syl6eqr 2662 . . . . . . . . 9 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = 𝐹)
127, 11sylan9eqr 2666 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑓 = 𝐹)
1312eleq1d 2672 . . . . . . 7 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
14 simpllr 795 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = (Base‘𝑔))
15 simplll 794 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑔 = 𝑊)
1615fveq2d 6107 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = (Base‘𝑊))
17 isphl.v . . . . . . . . . 10 𝑉 = (Base‘𝑊)
1816, 17syl6eqr 2662 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = 𝑉)
1914, 18eqtrd 2644 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = 𝑉)
20 simplr 788 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = (·𝑖𝑔))
2115fveq2d 6107 . . . . . . . . . . . . . 14 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = (·𝑖𝑊))
22 isphl.h . . . . . . . . . . . . . 14 , = (·𝑖𝑊)
2321, 22syl6eqr 2662 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = , )
2420, 23eqtrd 2644 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = , )
2524oveqd 6566 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑥) = (𝑦 , 𝑥))
2619, 25mpteq12dv 4663 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑣 ↦ (𝑦𝑥)) = (𝑦𝑉 ↦ (𝑦 , 𝑥)))
2712fveq2d 6107 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (ringLMod‘𝑓) = (ringLMod‘𝐹))
2815, 27oveq12d 6567 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑔 LMHom (ringLMod‘𝑓)) = (𝑊 LMHom (ringLMod‘𝐹)))
2926, 28eleq12d 2682 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ↔ (𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹))))
3024oveqd 6566 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑥) = (𝑥 , 𝑥))
3112fveq2d 6107 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = (0g𝐹))
32 isphl.z . . . . . . . . . . . 12 𝑍 = (0g𝐹)
3331, 32syl6eqr 2662 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = 𝑍)
3430, 33eqeq12d 2625 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑥𝑥) = (0g𝑓) ↔ (𝑥 , 𝑥) = 𝑍))
3515fveq2d 6107 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = (0g𝑊))
36 isphl.o . . . . . . . . . . . 12 0 = (0g𝑊)
3735, 36syl6eqr 2662 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = 0 )
3837eqeq2d 2620 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥 = (0g𝑔) ↔ 𝑥 = 0 ))
3934, 38imbi12d 333 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ↔ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 )))
4012fveq2d 6107 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = (*𝑟𝐹))
41 isphl.i . . . . . . . . . . . . 13 = (*𝑟𝐹)
4240, 41syl6eqr 2662 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = )
4324oveqd 6566 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑦) = (𝑥 , 𝑦))
4442, 43fveq12d 6109 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((*𝑟𝑓)‘(𝑥𝑦)) = ( ‘(𝑥 , 𝑦)))
4544, 25eqeq12d 2625 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4619, 45raleqbidv 3129 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4729, 39, 463anbi123d 1391 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)) ↔ ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
4819, 47raleqbidv 3129 . . . . . . 7 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)) ↔ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
4913, 48anbi12d 743 . . . . . 6 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
506, 49sbcied 3439 . . . . 5 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → ([(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
514, 50sbcied 3439 . . . 4 ((𝑔 = 𝑊𝑣 = (Base‘𝑔)) → ([(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
522, 51sbcied 3439 . . 3 (𝑔 = 𝑊 → ([(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥))) ↔ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
53 df-phl 19790 . . 3 PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}
5452, 53elrab2 3333 . 2 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
55 3anass 1035 . 2 ((𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))) ↔ (𝑊 ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
5654, 55bitr4i 266 1 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  *𝑟cstv 15770  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  *-Ringcsr 18667   LMHom clmhm 18840  LVecclvec 18923  ringLModcrglmod 18990  PreHilcphl 19788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-phl 19790 This theorem is referenced by:  phllvec  19793  phlsrng  19795  phllmhm  19796  ipcj  19798  ipeq0  19802  isphld  19818  phlpropd  19819
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