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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdihjatcc 30301 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjat 30302 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrnlem1N 30303 Lemma for dihprrn 30305, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

Theoremdihprrnlem2 30304 Lemma for dihprrn 30305. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrn 30305 The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)

Theoremdjhlsmat 30306 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 30305; should we directly use dihjat 30302? (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdihjat1lem 30307 Subspace sum of a closed subspace and an atom. (pmapjat1 28731 analog.) TODO: merge into dihjat1 30308? (Contributed by NM, 18-Aug-2014.)
joinH

Theoremdihjat1 30308 Subspace sum of a closed subspace and an atom. (pmapjat1 28731 analog.) (Contributed by NM, 1-Oct-2014.)
joinH

Theoremdihsmsprn 30309 Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)

Theoremdihjat2 30310 The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
joinH                     LSAtoms

Theoremdihjat3 30311 Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)

Theoremdihjat4 30312 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihjat6 30313 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihsmsnrn 30314 The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)

Theoremdihsmatrn 30315 The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 30310. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdihjat5N 30316 Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)

Theoremdvh4dimat 30317* There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh3dimatN 30318* There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh2dimatN 30319* Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh1dimat 30320* There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh1dim 30321* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)

Theoremdvh4dimlem 30322* Lemma for dvh4dimN 30326. (Contributed by NM, 22-May-2015.)

Theoremdvhdimlem 30323* Lemma for dvh2dim 30324 and dvh3dim 30325. TODO: make this obsolete and use dvh4dimlem 30322 directly? (Contributed by NM, 24-Apr-2015.)

Theoremdvh2dim 30324* There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)

Theoremdvh3dim 30325* There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)

Theoremdvh4dimN 30326* There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)

Theoremdvh3dim2 30327* There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)

Theoremdvh3dim3N 30328* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 30327 everywhere. If this one is needed, make dvh3dim2 30327 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)

Theoremdochsnnz 30329 The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)

Theoremdochsatshp 30330 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsatshpb 30331 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsnshp 30332 The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp

Theoremdochshpsat 30333 A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochkrsat 30334 The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms       LFnl       LKer

Theoremdochkrsat2 30335 The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochsat0 30336 The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochkrsm 30337 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 30293 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl       LKer

Theoremdochexmidat 30338 Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)

Theoremdochexmidlem1 30339 Lemma for dochexmid 30347. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem2 30340 Lemma for dochexmid 30347. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem3 30341 Lemma for dochexmid 30347. Use atom exchange lsatexch1 27925 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem4 30342 Lemma for dochexmid 30347. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem5 30343 Lemma for dochexmid 30347. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem6 30344 Lemma for dochexmid 30347. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem7 30345 Lemma for dochexmid 30347. Contradict dochexmidlem6 30344. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem8 30346 Lemma for dochexmid 30347. The contradiction of dochexmidlem6 30344 and dochexmidlem7 30345 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmid 30347 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 30256. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 28856 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochsnkrlem1 30348 Lemma for dochsnkr 30351. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochsnkrlem2 30349 Lemma for dochsnkr 30351. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer                            LSAtoms

Theoremdochsnkrlem3 30350 Lemma for dochsnkr 30351. (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr 30351 A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems) (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr2 30352* Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 27996. (Contributed by NM, 27-Oct-2014.)
LKer       Scalar

Theoremdochsnkr2cl 30353* The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer       Scalar

Theoremdochflcl 30354* Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 27995. (Contributed by NM, 27-Oct-2014.)
LFnl       Scalar

Theoremdochfl1 30355* The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar

Theoremdochfln0 30356 The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1 30357* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 27949. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1OLDN 30358* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 27949. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer

16.22.11  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 30359 Extend class notation with all polarities of a left module or left vector space.
LPol

Definitiondf-lpolN 30360* Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
LPol LSAtoms LSHyp

TheoremlpolsetN 30361* The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremislpolN 30362* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremislpoldN 30363* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremlpolfN 30364 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolvN 30365 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolconN 30366 Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolsatN 30367 The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremlpolpolsatN 30368 Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LPol

TheoremdochpolN 30369 The subspace orthocomplement for the vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
LPol

Theoremlcfl1lem 30370* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)

Theoremlcfl1 30371* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)

Theoremlcfl2 30372* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlcfl3 30373* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremlcfl4N 30374* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
LSHyp       LFnl       LKer

Theoremlcfl5 30375* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlcfl5a 30376 Property of a functional with a closed kernel. TODO: Make lcfl5 30375 etc. obsolete and rewrite w/out hypothesis? (Contributed by NM, 29-Jan-2015.)
LFnl       LKer

Theoremlcfl6lem 30377* Lemma for lcfl6 30379. A functional (whose kernel is closed by dochsnkr 30351) is comletely determined by a vector in the orthocomplement in its kernel at which the functional value is 1. Note that the in the hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
Scalar                            LFnl       LKer

Theoremlcfl7lem 30378* Lemma for lcfl7N 30380. If two functionals and are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
Scalar                     LFnl       LKer

Theoremlcfl6 30379* Property of a functional with a closed kernel. Note that means the functional is zero by lkr0f 27973. (Contributed by NM, 3-Jan-2015.)
Scalar                     LFnl       LKer

Theoremlcfl7N 30380* Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that means the functional is zero by lkr0f 27973. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer

Theoremlcfl8 30381* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
LFnl       LKer

Theoremlcfl8a 30382* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
LFnl       LKer

Theoremlcfl8b 30383* Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
LFnl       LKer       LDual

Theoremlcfl9a 30384 Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
LFnl       LKer

Theoremlclkrlem1 30385* The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
LFnl       LKer       LDual       Scalar

Theoremlclkrlem2a 30386 Lemma for lclkr 30412. Use lshpat 27935 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)
LSAtoms

Theoremlclkrlem2b 30387 Lemma for lclkr 30412. (Contributed by NM, 17-Jan-2015.)
LSAtoms

Theoremlclkrlem2c 30388 Lemma for lclkr 30412. (Contributed by NM, 16-Jan-2015.)
LSAtoms                                                 LSHyp

Theoremlclkrlem2d 30389 Lemma for lclkr 30412. (Contributed by NM, 16-Jan-2015.)
LSAtoms

Theoremlclkrlem2e 30390 Lemma for lclkr 30412. The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015.)
LFnl       LKer       LDual

Theoremlclkrlem2f 30391 Lemma for lclkr 30412. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2g 30392 Lemma for lclkr 30412. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2h 30393 Lemma for lclkr 30412. Eliminate the hypothesis. (Contributed by NM, 16-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2i 30394 Lemma for lclkr 30412. Eliminate the hypothesis. (Contributed by NM, 17-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2j 30395 Lemma for lclkr 30412. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2k 30396 Lemma for lclkr 30412. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2l 30397 Lemma for lclkr 30412. Eliminate the , hypotheses. (Contributed by NM, 18-Jan-2015.)
Scalar                                   LFnl       LSHyp       LKer       LDual

Theoremlclkrlem2m 30398 Lemma for lclkr 30412. Construct a vector that makes the sum of functionals zero. Combine with to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
Scalar                                   LFnl       LDual

Theoremlclkrlem2n 30399 Lemma for lclkr 30412. (Contributed by NM, 12-Jan-2015.)
Scalar                                   LFnl       LDual                                                 LKer

Theoremlclkrlem2o 30400 Lemma for lclkr 30412. When is nonzero, the vectors and can't both belong to the hyperplane generated by . (Contributed by NM, 17-Jan-2015.)
Scalar                                   LFnl       LDual                                                 LKer

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