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

Theoremcdlemn11a 30301* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 30302* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 30303* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 30304* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 30301, cdlemn11b 30302, cdlemn11c 30303, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 30305 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 30306 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 30307* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 30308* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 30309* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 30310 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

Theoremdihjust 30311 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

Theoremdihord1 30312 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change to using lhpmcvr3 29118, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Theoremdihord2a 30313 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2b 30314 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2cN 30315* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)

Theoremdihord11b 30316* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord10 30317* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord11c 30318* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre 30319* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre2 30320* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)

Theoremdihord2 30321 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. Todo: do we need and ? (Contributed by NM, 4-Mar-2014.)

Syntaxcdih 30322 Extend class notation with isomorphism H.

Definitiondf-dih 30323* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)

Theoremdihffval 30324* The isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihfval 30325* Isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihval 30326* Value of isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)

Theoremdihvalc 30327* Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

Theoremdihlsscpre 30328 Closure of isomorphism H for a lattice when . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcqpre 30329 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcq 30330 Value of isomorphism H for a lattice when , given auxiliary atom . TODO: Use dihvalcq2 30341 (with lhpmcvr3 29118 for simplification) that changes and to and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)

Theoremdihvalb 30331 Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

TheoremdihopelvalbN 30332* Ordered pair member of the partial isomorphism H for argument under . (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihvalcqat 30333 Value of isomorphism H for a lattice at an atom not under . (Contributed by NM, 27-Mar-2014.)

Theoremdih1dimb 30334* Two expressions for a 1-dimensional subspace of vector space H (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimb2 30335* Isomorphism H at an atom under . (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimc 30336* Isomorphism H at an atom not under . (Contributed by NM, 27-Apr-2014.)

Theoremdib2dim 30337 Extend dia2dim 30171 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimb 30338 Extend dib2dim 30337 to isomorphism H. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimbALTN 30339 Extend dia2dim 30171 to isomorphism H. (This version combines dib2dim 30337 and dih2dimb 30338 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdihopelvalcqat 30340* Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)

Theoremdihvalcq2 30341 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.)

Theoremdihopelvalcpre 30342* Member of value of isomorphism H for a lattice when , given auxiliary atom . TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)

Theoremdihopelvalc 30343* Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.)

Theoremdihlss 30344 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)

Theoremdihss 30345 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihssxp 30346 An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.)

Theoremdihopcl 30347 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)

TheoremxihopellsmN 30348* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

Theoremdihopellsm 30349* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)

Theoremdihord6apre 30350* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord3 30351 The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihord4 30352 The isomorphism H for a lattice is order-preserving in the region not under co-atom . TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-Mar-2014.)

Theoremdihord5b 30353 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (Contributed by NM, 7-Mar-2014.)

Theoremdihord6b 30354 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord6a 30355 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5apre 30356 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5a 30357 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord 30358 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdih11 30359 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihf11lem 30360 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)

Theoremdihf11 30361 The isomorphism H for a lattice is a one-to-one function. . Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihfn 30362 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihdm 30363 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihcl 30364 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvcl 30365 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvid1 30366 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvid2 30367 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvord 30368 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)

Theoremdihcnv11 30369 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)

Theoremdihsslss 30370 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnlss 30371 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnss 30372 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihvalrel 30373 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)

Theoremdih0 30374 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)

Theoremdih0bN 30375 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0vbN 30376 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0cnv 30377 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)

Theoremdih0rn 30378 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

Theoremdih0sb 30379 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)

Theoremdih1 30380 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Theoremdih1rn 30381 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)

Theoremdih1cnv 30382 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)

TheoremdihwN 30383* Value of isomorphism H at the fiducial hyperplane . (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Theoremdihmeetlem1N 30384* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5apreN 30385* A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem5aN 30386 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2aN 30387* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem2N 30388* The GLB of a set of lattice elements is the same as that of the set with elements of cut down to be under . (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3N 30389* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

Theoremdihglblem3aN 30390* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdihglblem4 30391* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

Theoremdihglblem5 30392* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

Theoremdihmeetlem2N 30393 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcpreN 30394* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)

TheoremdihglbcN 30395* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetcN 30396 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbN 30397 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane . (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)

TheoremdihmeetbclemN 30398 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem3N 30399 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Theoremdihmeetlem4preN 30400* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

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