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Theorem isphl 20190
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 fvexd 6346 . . . 4 (𝑔 = 𝑊 → (Base‘𝑔) ∈ V)
2 fvexd 6346 . . . . 5 ((𝑔 = 𝑊𝑣 = (Base‘𝑔)) → (·𝑖𝑔) ∈ V)
3 fvexd 6346 . . . . . 6 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) ∈ V)
4 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑔) → 𝑓 = (Scalar‘𝑔))
5 simpll 750 . . . . . . . . . . 11 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → 𝑔 = 𝑊)
65fveq2d 6337 . . . . . . . . . 10 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = (Scalar‘𝑊))
7 isphl.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
86, 7syl6eqr 2823 . . . . . . . . 9 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = 𝐹)
94, 8sylan9eqr 2827 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑓 = 𝐹)
109eleq1d 2835 . . . . . . 7 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
11 simpllr 760 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = (Base‘𝑔))
12 simplll 758 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑔 = 𝑊)
1312fveq2d 6337 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = (Base‘𝑊))
14 isphl.v . . . . . . . . . 10 𝑉 = (Base‘𝑊)
1513, 14syl6eqr 2823 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = 𝑉)
1611, 15eqtrd 2805 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = 𝑉)
17 simplr 752 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = (·𝑖𝑔))
1812fveq2d 6337 . . . . . . . . . . . . . 14 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = (·𝑖𝑊))
19 isphl.h . . . . . . . . . . . . . 14 , = (·𝑖𝑊)
2018, 19syl6eqr 2823 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = , )
2117, 20eqtrd 2805 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = , )
2221oveqd 6813 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑥) = (𝑦 , 𝑥))
2316, 22mpteq12dv 4868 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑣 ↦ (𝑦𝑥)) = (𝑦𝑉 ↦ (𝑦 , 𝑥)))
249fveq2d 6337 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (ringLMod‘𝑓) = (ringLMod‘𝐹))
2512, 24oveq12d 6814 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑔 LMHom (ringLMod‘𝑓)) = (𝑊 LMHom (ringLMod‘𝐹)))
2623, 25eleq12d 2844 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ↔ (𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹))))
2721oveqd 6813 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑥) = (𝑥 , 𝑥))
289fveq2d 6337 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = (0g𝐹))
29 isphl.z . . . . . . . . . . . 12 𝑍 = (0g𝐹)
3028, 29syl6eqr 2823 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = 𝑍)
3127, 30eqeq12d 2786 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑥𝑥) = (0g𝑓) ↔ (𝑥 , 𝑥) = 𝑍))
3212fveq2d 6337 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = (0g𝑊))
33 isphl.o . . . . . . . . . . . 12 0 = (0g𝑊)
3432, 33syl6eqr 2823 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = 0 )
3534eqeq2d 2781 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥 = (0g𝑔) ↔ 𝑥 = 0 ))
3631, 35imbi12d 333 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ↔ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 )))
379fveq2d 6337 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = (*𝑟𝐹))
38 isphl.i . . . . . . . . . . . . 13 = (*𝑟𝐹)
3937, 38syl6eqr 2823 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = )
4021oveqd 6813 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑦) = (𝑥 , 𝑦))
4139, 40fveq12d 6340 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((*𝑟𝑓)‘(𝑥𝑦)) = ( ‘(𝑥 , 𝑦)))
4241, 22eqeq12d 2786 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4316, 42raleqbidv 3301 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4426, 36, 433anbi123d 1547 . . . . . . . 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))))
4516, 44raleqbidv 3301 . . . . . . 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))))
4610, 45anbi12d 616 . . . . . 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)))))
473, 46sbcied 3624 . . . . 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)))))
482, 47sbcied 3624 . . . 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)))))
491, 48sbcied 3624 . . 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)))))
50 df-phl 20188 . . 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)))}
5149, 50elrab2 3518 . 2 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))))
52 3anass 1080 . 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)))))
5351, 52bitr4i 267 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  [wsbc 3587  cmpt 4864  cfv 6030  (class class class)co 6796  Basecbs 16064  *𝑟cstv 16151  Scalarcsca 16152  ·𝑖cip 16154  0gc0g 16308  *-Ringcsr 19054   LMHom clmhm 19232  LVecclvec 19315  ringLModcrglmod 19384  PreHilcphl 20186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-iota 5993  df-fv 6038  df-ov 6799  df-phl 20188
This theorem is referenced by:  phllvec  20191  phlsrng  20193  phllmhm  20194  ipcj  20196  ipeq0  20200  isphld  20216  phlpropd  20217
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