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Theorem isphl 19913
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 6170 . . . 4 (𝑔 = 𝑊 → (Base‘𝑔) ∈ V)
2 fvexd 6170 . . . . 5 ((𝑔 = 𝑊𝑣 = (Base‘𝑔)) → (·𝑖𝑔) ∈ V)
3 fvexd 6170 . . . . . 6 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) ∈ V)
4 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑔) → 𝑓 = (Scalar‘𝑔))
5 simpll 789 . . . . . . . . . . 11 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → 𝑔 = 𝑊)
65fveq2d 6162 . . . . . . . . . 10 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = (Scalar‘𝑊))
7 isphl.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
86, 7syl6eqr 2673 . . . . . . . . 9 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = 𝐹)
94, 8sylan9eqr 2677 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑓 = 𝐹)
109eleq1d 2683 . . . . . . 7 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
11 simpllr 798 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = (Base‘𝑔))
12 simplll 797 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑔 = 𝑊)
1312fveq2d 6162 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = (Base‘𝑊))
14 isphl.v . . . . . . . . . 10 𝑉 = (Base‘𝑊)
1513, 14syl6eqr 2673 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = 𝑉)
1611, 15eqtrd 2655 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = 𝑉)
17 simplr 791 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = (·𝑖𝑔))
1812fveq2d 6162 . . . . . . . . . . . . . 14 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = (·𝑖𝑊))
19 isphl.h . . . . . . . . . . . . . 14 , = (·𝑖𝑊)
2018, 19syl6eqr 2673 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = , )
2117, 20eqtrd 2655 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = , )
2221oveqd 6632 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑥) = (𝑦 , 𝑥))
2316, 22mpteq12dv 4703 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑣 ↦ (𝑦𝑥)) = (𝑦𝑉 ↦ (𝑦 , 𝑥)))
249fveq2d 6162 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (ringLMod‘𝑓) = (ringLMod‘𝐹))
2512, 24oveq12d 6633 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑔 LMHom (ringLMod‘𝑓)) = (𝑊 LMHom (ringLMod‘𝐹)))
2623, 25eleq12d 2692 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ↔ (𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹))))
2721oveqd 6632 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑥) = (𝑥 , 𝑥))
289fveq2d 6162 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = (0g𝐹))
29 isphl.z . . . . . . . . . . . 12 𝑍 = (0g𝐹)
3028, 29syl6eqr 2673 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = 𝑍)
3127, 30eqeq12d 2636 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑥𝑥) = (0g𝑓) ↔ (𝑥 , 𝑥) = 𝑍))
3212fveq2d 6162 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = (0g𝑊))
33 isphl.o . . . . . . . . . . . 12 0 = (0g𝑊)
3432, 33syl6eqr 2673 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = 0 )
3534eqeq2d 2631 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥 = (0g𝑔) ↔ 𝑥 = 0 ))
3631, 35imbi12d 334 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ↔ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 )))
379fveq2d 6162 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = (*𝑟𝐹))
38 isphl.i . . . . . . . . . . . . 13 = (*𝑟𝐹)
3937, 38syl6eqr 2673 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = )
4021oveqd 6632 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑦) = (𝑥 , 𝑦))
4139, 40fveq12d 6164 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((*𝑟𝑓)‘(𝑥𝑦)) = ( ‘(𝑥 , 𝑦)))
4241, 22eqeq12d 2636 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4316, 42raleqbidv 3145 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4426, 36, 433anbi123d 1396 . . . . . . . 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 3145 . . . . . . 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 746 . . . . . 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 3459 . . . . 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 3459 . . . 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 3459 . . 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 19911 . . 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 3353 . 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 1040 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  [wsbc 3422  cmpt 4683  cfv 5857  (class class class)co 6615  Basecbs 15800  *𝑟cstv 15883  Scalarcsca 15884  ·𝑖cip 15886  0gc0g 16040  *-Ringcsr 18784   LMHom clmhm 18959  LVecclvec 19042  ringLModcrglmod 19109  PreHilcphl 19909
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-iota 5820  df-fv 5865  df-ov 6618  df-phl 19911
This theorem is referenced by:  phllvec  19914  phlsrng  19916  phllmhm  19917  ipcj  19919  ipeq0  19923  isphld  19939  phlpropd  19940
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