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Theorem isphl 21180
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 = (Baseβ€˜π‘Š)
isphl.f 𝐹 = (Scalarβ€˜π‘Š)
isphl.h , = (Β·π‘–β€˜π‘Š)
isphl.o 0 = (0gβ€˜π‘Š)
isphl.i βˆ— = (*π‘Ÿβ€˜πΉ)
isphl.z 𝑍 = (0gβ€˜πΉ)
Assertion
Ref Expression
isphl (π‘Š ∈ PreHil ↔ (π‘Š ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯))))
Distinct variable groups:   π‘₯,𝑦,𝑉   π‘₯,π‘Š,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)   , (π‘₯,𝑦)   βˆ— (π‘₯,𝑦)   0 (π‘₯,𝑦)   𝑍(π‘₯,𝑦)
Proof of Theorem isphl
Dummy variables 𝑓 𝑔 β„Ž 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6906 . . . 4 (𝑔 = π‘Š β†’ (Baseβ€˜π‘”) ∈ V)
2 fvexd 6906 . . . . 5 ((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) β†’ (Β·π‘–β€˜π‘”) ∈ V)
3 fvexd 6906 . . . . . 6 (((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) β†’ (Scalarβ€˜π‘”) ∈ V)
4 id 22 . . . . . . . . 9 (𝑓 = (Scalarβ€˜π‘”) β†’ 𝑓 = (Scalarβ€˜π‘”))
5 simpll 765 . . . . . . . . . . 11 (((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) β†’ 𝑔 = π‘Š)
65fveq2d 6895 . . . . . . . . . 10 (((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) β†’ (Scalarβ€˜π‘”) = (Scalarβ€˜π‘Š))
7 isphl.f . . . . . . . . . 10 𝐹 = (Scalarβ€˜π‘Š)
86, 7eqtr4di 2790 . . . . . . . . 9 (((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) β†’ (Scalarβ€˜π‘”) = 𝐹)
94, 8sylan9eqr 2794 . . . . . . . 8 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ 𝑓 = 𝐹)
109eleq1d 2818 . . . . . . 7 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (𝑓 ∈ *-Ring ↔ 𝐹 ∈ *-Ring))
11 simpllr 774 . . . . . . . . 9 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ 𝑣 = (Baseβ€˜π‘”))
12 simplll 773 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ 𝑔 = π‘Š)
1312fveq2d 6895 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (Baseβ€˜π‘”) = (Baseβ€˜π‘Š))
14 isphl.v . . . . . . . . . 10 𝑉 = (Baseβ€˜π‘Š)
1513, 14eqtr4di 2790 . . . . . . . . 9 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (Baseβ€˜π‘”) = 𝑉)
1611, 15eqtrd 2772 . . . . . . . 8 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ 𝑣 = 𝑉)
17 simplr 767 . . . . . . . . . . . . 13 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ β„Ž = (Β·π‘–β€˜π‘”))
1812fveq2d 6895 . . . . . . . . . . . . . 14 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (Β·π‘–β€˜π‘”) = (Β·π‘–β€˜π‘Š))
19 isphl.h . . . . . . . . . . . . . 14 , = (Β·π‘–β€˜π‘Š)
2018, 19eqtr4di 2790 . . . . . . . . . . . . 13 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (Β·π‘–β€˜π‘”) = , )
2117, 20eqtrd 2772 . . . . . . . . . . . 12 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ β„Ž = , )
2221oveqd 7425 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (π‘¦β„Žπ‘₯) = (𝑦 , π‘₯))
2316, 22mpteq12dv 5239 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) = (𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)))
249fveq2d 6895 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (ringLModβ€˜π‘“) = (ringLModβ€˜πΉ))
2512, 24oveq12d 7426 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (𝑔 LMHom (ringLModβ€˜π‘“)) = (π‘Š LMHom (ringLModβ€˜πΉ)))
2623, 25eleq12d 2827 . . . . . . . . 9 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ↔ (𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ))))
2721oveqd 7425 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (π‘₯β„Žπ‘₯) = (π‘₯ , π‘₯))
289fveq2d 6895 . . . . . . . . . . . 12 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (0gβ€˜π‘“) = (0gβ€˜πΉ))
29 isphl.z . . . . . . . . . . . 12 𝑍 = (0gβ€˜πΉ)
3028, 29eqtr4di 2790 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (0gβ€˜π‘“) = 𝑍)
3127, 30eqeq12d 2748 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) ↔ (π‘₯ , π‘₯) = 𝑍))
3212fveq2d 6895 . . . . . . . . . . . 12 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (0gβ€˜π‘”) = (0gβ€˜π‘Š))
33 isphl.o . . . . . . . . . . . 12 0 = (0gβ€˜π‘Š)
3432, 33eqtr4di 2790 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (0gβ€˜π‘”) = 0 )
3534eqeq2d 2743 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (π‘₯ = (0gβ€˜π‘”) ↔ π‘₯ = 0 ))
3631, 35imbi12d 344 . . . . . . . . 9 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ↔ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 )))
379fveq2d 6895 . . . . . . . . . . . . 13 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (*π‘Ÿβ€˜π‘“) = (*π‘Ÿβ€˜πΉ))
38 isphl.i . . . . . . . . . . . . 13 βˆ— = (*π‘Ÿβ€˜πΉ)
3937, 38eqtr4di 2790 . . . . . . . . . . . 12 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (*π‘Ÿβ€˜π‘“) = βˆ— )
4021oveqd 7425 . . . . . . . . . . . 12 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (π‘₯β„Žπ‘¦) = (π‘₯ , 𝑦))
4139, 40fveq12d 6898 . . . . . . . . . . 11 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = ( βˆ— β€˜(π‘₯ , 𝑦)))
4241, 22eqeq12d 2748 . . . . . . . . . 10 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯) ↔ ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))
4316, 42raleqbidv 3342 . . . . . . . . 9 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯) ↔ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))
4426, 36, 433anbi123d 1436 . . . . . . . 8 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯)) ↔ ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯))))
4516, 44raleqbidv 3342 . . . . . . 7 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ (βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯)) ↔ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯))))
4610, 45anbi12d 631 . . . . . 6 ((((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) ∧ 𝑓 = (Scalarβ€˜π‘”)) β†’ ((𝑓 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯))) ↔ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
473, 46sbcied 3822 . . . . 5 (((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) ∧ β„Ž = (Β·π‘–β€˜π‘”)) β†’ ([(Scalarβ€˜π‘”) / 𝑓](𝑓 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯))) ↔ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
482, 47sbcied 3822 . . . 4 ((𝑔 = π‘Š ∧ 𝑣 = (Baseβ€˜π‘”)) β†’ ([(Β·π‘–β€˜π‘”) / β„Ž][(Scalarβ€˜π‘”) / 𝑓](𝑓 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯))) ↔ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
491, 48sbcied 3822 . . 3 (𝑔 = π‘Š β†’ ([(Baseβ€˜π‘”) / 𝑣][(Β·π‘–β€˜π‘”) / β„Ž][(Scalarβ€˜π‘”) / 𝑓](𝑓 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯))) ↔ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
50 df-phl 21178 . . 3 PreHil = {𝑔 ∈ LVec ∣ [(Baseβ€˜π‘”) / 𝑣][(Β·π‘–β€˜π‘”) / β„Ž][(Scalarβ€˜π‘”) / 𝑓](𝑓 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (π‘¦β„Žπ‘₯)) ∈ (𝑔 LMHom (ringLModβ€˜π‘“)) ∧ ((π‘₯β„Žπ‘₯) = (0gβ€˜π‘“) β†’ π‘₯ = (0gβ€˜π‘”)) ∧ βˆ€π‘¦ ∈ 𝑣 ((*π‘Ÿβ€˜π‘“)β€˜(π‘₯β„Žπ‘¦)) = (π‘¦β„Žπ‘₯)))}
5149, 50elrab2 3686 . 2 (π‘Š ∈ PreHil ↔ (π‘Š ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
52 3anass 1095 . 2 ((π‘Š ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯))) ↔ (π‘Š ∈ LVec ∧ (𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯)))))
5351, 52bitr4i 277 1 (π‘Š ∈ PreHil ↔ (π‘Š ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ βˆ€π‘₯ ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((π‘₯ , π‘₯) = 𝑍 β†’ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 ( βˆ— β€˜(π‘₯ , 𝑦)) = (𝑦 , π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474  [wsbc 3777   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  *π‘Ÿcstv 17198  Scalarcsca 17199  Β·π‘–cip 17201  0gc0g 17384  *-Ringcsr 20451   LMHom clmhm 20629  LVecclvec 20712  ringLModcrglmod 20781  PreHilcphl 21176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6495  df-fv 6551  df-ov 7411  df-phl 21178
This theorem is referenced by:  phllvec  21181  phlsrng  21183  phllmhm  21184  ipcj  21186  ipeq0  21190  isphld  21206  phlpropd  21207
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