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Theorem List for Metamath Proof Explorer - 32401-32447   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhdmapip0 32401 Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)
Scalar              HDMap

Theoremhdmapip1 32402 Construct a proportional vector whose inner product with the original equals one. (Contributed by NM, 13-Jun-2015.)
Scalar                     HDMap

Theoremhdmapip0com 32403 Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.)
Scalar              HDMap

Theoremhdmapinvlem1 32404 Line 27 in [Baer] p. 110. We use for Baer's u. Our unit vector has the required properties for his w by hdmapevec2 32322. Our means the inner product i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.)
Scalar                            HDMap

Theoremhdmapinvlem2 32405 Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.)
Scalar                            HDMap

Theoremhdmapinvlem3 32406 Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhdmapinvlem4 32407 Part 1.1 of Proposition 1 of [Baer] p. 110. We use , , , and for Baer's u, v, s, and t. Our unit vector has the required properties for his w by hdmapevec2 32322. Our means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhdmapglem5 32408 Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhgmapvvlem1 32409 Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our , , , , correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvvlem2 32410 Lemma for hgmapvv 32412. Eliminate (Baer's s). (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvvlem3 32411 Lemma for hgmapvv 32412. Eliminate (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvv 32412 Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.)
Scalar              HGMap

Theoremhdmapglem7a 32413* Lemma for hdmapg 32416. (Contributed by NM, 14-Jun-2015.)
Scalar

Theoremhdmapglem7b 32414 Lemma for hdmapg 32416. (Contributed by NM, 14-Jun-2015.)
Scalar                                                               HDMap       HGMap

Theoremhdmapglem7 32415 Lemma for hdmapg 32416. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our , , , , , , correspond to Baer's w, H, x, y, x', x'', y' , y'', and our corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
Scalar                                                               HDMap       HGMap

Theoremhdmapg 32416 Apply the scalar sigma function (involution) to an inner product reverses the arguments. The inner product of and is represented by . Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.)
HDMap       HGMap

Theoremhdmapoc 32417* Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)
Scalar                     HDMap

Syntaxchlh 32418 Extend class notation with the final constructed Hilbert space.
HLHil

Definitiondf-hlhil 32419* Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil Scalar sSet HGMap HDMap

Theoremhlhilset 32420* The final Hilbert space constructed from a Hilbert lattice and an arbitrary hyperplane in . (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil                                   HGMap       sSet               HDMap                     Scalar

Theoremhlhilsca 32421 The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil                     HGMap       sSet        Scalar

Theoremhlhilbase 32422 The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil

Theoremhlhilplus 32423 The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.)
HLHil

Theoremhlhilslem 32424 Lemma for hlhilsbase2 32428. (Contributed by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar              Slot

Theoremhlhilsbase 32425 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsplus 32426 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsmul 32427 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsbase2 32428 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhilsplus2 32429 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhilsmul2 32430 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhils0 32431 The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhils1N 32432 The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.)
Scalar       HLHil       Scalar

Theoremhlhilvsca 32433 The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil

Theoremhlhilip 32434* Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HDMap       HLHil

Theoremhlhilipval 32435 Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HDMap       HLHil

Theoremhlhilnvl 32436 The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar       HGMap

Theoremhlhillvec 32437 The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhildrng 32438 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil              Scalar

Theoremhlhilsrnglem 32439 Lemma for hlhilsrng 32440. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil              Scalar              Scalar                            HGMap

Theoremhlhilsrng 32440 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.)
HLHil              Scalar

Theoremhlhil0 32441 The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhillsm 32442 The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhilocv 32443 The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhillcs 32444 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 32422 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.)
HLHil

Theoremhlhilphllem 32445* Lemma for hlhil 19297. (Contributed by NM, 23-Jun-2015.)
HLHil              Scalar                                   Scalar                                                 HDMap       HGMap

Theoremhlhilhillem 32446* Lemma for hlhil 19297. (Contributed by NM, 23-Jun-2015.)
HLHil              Scalar                                   Scalar                                                 HDMap       HGMap

Theoremhlathil 32447 Construction of a Hilbert space (df-hil 16886) from a Hilbert lattice (df-hlat 29834) , where is a fixed but arbitrary hyperplane (co-atom) in .

The Hilbert space is identical to the vector space (see dvhlvec 31592) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely.

An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to . See additional discussion at http://us.metamath.org/qlegif/mmql.html#what.

corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a always exists since has lattice rank of at least 4 by df-hil 16886. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.)

HLHil

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