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Theorem List for Metamath Proof Explorer - 31301-31400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdicssdvh 31301 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)

Theoremdicelval1sta 31302* Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicelval1stN 31303 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)

Theoremdicelval2nd 31304 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicvaddcl 31305 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Theoremdicvscacl 31306 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremdicn0 31307 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Theoremdiclss 31308 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Theoremdiclspsn 31309* The value of isomorphism C is spanned by vector . Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn2 31310* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn2a 31311* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn3 31312* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn4 31313* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn4a 31314* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn5pre 31315* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn5 31316 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn6 31317* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn7 31318* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn8 31319* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn9 31320* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn10 31321 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11a 31322* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 31323* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 31324* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 31325* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 31322, cdlemn11b 31323, cdlemn11c 31324, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 31326 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 31327 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 31328* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 31329* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 31330* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 31331 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 31332 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 31333 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change to using lhpmcvr3 30139, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Theoremdihord2a 31334 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2b 31335 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2cN 31336* 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 31337* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord10 31338* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord11c 31339* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre 31340* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre2 31341* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)

Theoremdihord2 31342 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 31343 Extend class notation with isomorphism H.

Definitiondf-dih 31344* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)

Theoremdihffval 31345* The isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihfval 31346* Isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihval 31347* Value of isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)

Theoremdihvalc 31348* Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

Theoremdihlsscpre 31349 Closure of isomorphism H for a lattice when . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcqpre 31350 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcq 31351 Value of isomorphism H for a lattice when , given auxiliary atom . TODO: Use dihvalcq2 31362 (with lhpmcvr3 30139 for simplification) that changes and to and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)

Theoremdihvalb 31352 Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

TheoremdihopelvalbN 31353* Ordered pair member of the partial isomorphism H for argument under . (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihvalcqat 31354 Value of isomorphism H for a lattice at an atom not under . (Contributed by NM, 27-Mar-2014.)

Theoremdih1dimb 31355* 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 31356* Isomorphism H at an atom under . (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimc 31357* Isomorphism H at an atom not under . (Contributed by NM, 27-Apr-2014.)

Theoremdib2dim 31358 Extend dia2dim 31192 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimb 31359 Extend dib2dim 31358 to isomorphism H. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimbALTN 31360 Extend dia2dim 31192 to isomorphism H. (This version combines dib2dim 31358 and dih2dimb 31359 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 31361* Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)

Theoremdihvalcq2 31362 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.)

Theoremdihopelvalcpre 31363* 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 31364* Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.)

Theoremdihlss 31365 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)

Theoremdihss 31366 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihssxp 31367 An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.)

Theoremdihopcl 31368 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)

TheoremxihopellsmN 31369* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

Theoremdihopellsm 31370* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)

Theoremdihord6apre 31371* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord3 31372 The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihord4 31373 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 31374 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 31375 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord6a 31376 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5apre 31377 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5a 31378 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord 31379 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 31380 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 31381 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)

Theoremdihf11 31382 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 31383 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihdm 31384 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihcl 31385 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvcl 31386 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvid1 31387 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvid2 31388 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvord 31389 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)

Theoremdihcnv11 31390 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)

Theoremdihsslss 31391 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnlss 31392 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnss 31393 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihvalrel 31394 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)

Theoremdih0 31395 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 31396 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0vbN 31397 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0cnv 31398 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)

Theoremdih0rn 31399 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

Theoremdih0sb 31400 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)

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