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Theorem isphl 21577
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 6911 . . . 4 (𝑔 = 𝑊 → (Base‘𝑔) ∈ V)
2 fvexd 6911 . . . . 5 ((𝑔 = 𝑊𝑣 = (Base‘𝑔)) → (·𝑖𝑔) ∈ V)
3 fvexd 6911 . . . . . 6 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) ∈ V)
4 id 22 . . . . . . . . 9 (𝑓 = (Scalar‘𝑔) → 𝑓 = (Scalar‘𝑔))
5 simpll 765 . . . . . . . . . . 11 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → 𝑔 = 𝑊)
65fveq2d 6900 . . . . . . . . . 10 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = (Scalar‘𝑊))
7 isphl.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
86, 7eqtr4di 2783 . . . . . . . . 9 (((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) → (Scalar‘𝑔) = 𝐹)
94, 8sylan9eqr 2787 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑓 = 𝐹)
109eleq1d 2810 . . . . . . 7 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
11 simpllr 774 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = (Base‘𝑔))
12 simplll 773 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑔 = 𝑊)
1312fveq2d 6900 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = (Base‘𝑊))
14 isphl.v . . . . . . . . . 10 𝑉 = (Base‘𝑊)
1513, 14eqtr4di 2783 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (Base‘𝑔) = 𝑉)
1611, 15eqtrd 2765 . . . . . . . 8 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → 𝑣 = 𝑉)
17 simplr 767 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = (·𝑖𝑔))
1812fveq2d 6900 . . . . . . . . . . . . . 14 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = (·𝑖𝑊))
19 isphl.h . . . . . . . . . . . . . 14 , = (·𝑖𝑊)
2018, 19eqtr4di 2783 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (·𝑖𝑔) = , )
2117, 20eqtrd 2765 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → = , )
2221oveqd 7436 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑥) = (𝑦 , 𝑥))
2316, 22mpteq12dv 5240 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑦𝑣 ↦ (𝑦𝑥)) = (𝑦𝑉 ↦ (𝑦 , 𝑥)))
249fveq2d 6900 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (ringLMod‘𝑓) = (ringLMod‘𝐹))
2512, 24oveq12d 7437 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑔 LMHom (ringLMod‘𝑓)) = (𝑊 LMHom (ringLMod‘𝐹)))
2623, 25eleq12d 2819 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ↔ (𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹))))
2721oveqd 7436 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑥) = (𝑥 , 𝑥))
289fveq2d 6900 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = (0g𝐹))
29 isphl.z . . . . . . . . . . . 12 𝑍 = (0g𝐹)
3028, 29eqtr4di 2783 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑓) = 𝑍)
3127, 30eqeq12d 2741 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((𝑥𝑥) = (0g𝑓) ↔ (𝑥 , 𝑥) = 𝑍))
3212fveq2d 6900 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = (0g𝑊))
33 isphl.o . . . . . . . . . . . 12 0 = (0g𝑊)
3432, 33eqtr4di 2783 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (0g𝑔) = 0 )
3534eqeq2d 2736 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥 = (0g𝑔) ↔ 𝑥 = 0 ))
3631, 35imbi12d 343 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ↔ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 )))
379fveq2d 6900 . . . . . . . . . . . . 13 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = (*𝑟𝐹))
38 isphl.i . . . . . . . . . . . . 13 = (*𝑟𝐹)
3937, 38eqtr4di 2783 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (*𝑟𝑓) = )
4021oveqd 7436 . . . . . . . . . . . 12 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (𝑥𝑦) = (𝑥 , 𝑦))
4139, 40fveq12d 6903 . . . . . . . . . . 11 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → ((*𝑟𝑓)‘(𝑥𝑦)) = ( ‘(𝑥 , 𝑦)))
4241, 22eqeq12d 2741 . . . . . . . . . 10 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4316, 42raleqbidv 3329 . . . . . . . . 9 ((((𝑔 = 𝑊𝑣 = (Base‘𝑔)) ∧ = (·𝑖𝑔)) ∧ 𝑓 = (Scalar‘𝑔)) → (∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥) ↔ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
4426, 36, 433anbi123d 1432 . . . . . . . 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 3329 . . . . . . 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 630 . . . . . 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 3819 . . . . 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 3819 . . . 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 3819 . . 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 21575 . . 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 3682 . 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 1092 . 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 277 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  [wsbc 3773  cmpt 5232  cfv 6549  (class class class)co 7419  Basecbs 17183  *𝑟cstv 17238  Scalarcsca 17239  ·𝑖cip 17241  0gc0g 17424  *-Ringcsr 20736   LMHom clmhm 20916  LVecclvec 20999  ringLModcrglmod 21069  PreHilcphl 21573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-iota 6501  df-fv 6557  df-ov 7422  df-phl 21575
This theorem is referenced by:  phllvec  21578  phlsrng  21580  phllmhm  21581  ipcj  21583  ipeq0  21587  isphld  21603  phlpropd  21604
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