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Theorem List for Metamath Proof Explorer - 32401-32500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcdval 32401* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual              LFnl       LKer       LDual              s

Theoremlcdval2 32402* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual              LFnl       LKer       LDual                     s

Theoremlcdlvec 32403 The dual vector space of functionals with closed kernels is a left vector space. (Contributed by NM, 14-Mar-2015.)
LCDual

Theoremlcdlmod 32404 The dual vector space of functionals with closed kernels is a left module. (Contributed by NM, 13-Mar-2015.)
LCDual

Theoremlcdvbase 32405* Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual                     LFnl       LKer

Theoremlcdvbasess 32406 The vector base set of the closed kernel dual space is a set of functionals. (Contributed by NM, 15-Mar-2015.)
LCDual                     LFnl

Theoremlcdvbaselfl 32407 A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.)
LCDual                     LFnl

Theoremlcdvbasecl 32408 Closure of the value of a vector (functional) in the closed kernel dual space. (Contributed by NM, 28-Mar-2015.)
Scalar              LCDual

Theoremlcdvadd 32409 Vector addition for the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
LDual              LCDual

Theoremlcdvaddval 32410 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
Scalar              LCDual

Theoremlcdsca 32411 The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Scalar       oppr       LCDual       Scalar

Theoremlcdsbase 32412 Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015.)
Scalar              LCDual       Scalar

Theoremlcdsadd 32413 Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual       Scalar

Theoremlcdsmul 32414 Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015.)
Scalar                     LCDual       Scalar

Theoremlcdvs 32415 Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
LDual              LCDual

Theoremlcdvsval 32416 Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
Scalar                     LCDual

Theoremlcdvscl 32417 The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual

Theoremlcdlssvscl 32418 Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual

Theoremlcdvsass 32419 Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
Scalar                     LCDual

Theoremlcd0 32420 The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual       Scalar

Theoremlcd1 32421 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual       Scalar

Theoremlcdneg 32422 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
Scalar              LCDual       Scalar

Theoremlcd0v 32423 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual

Theoremlcd0v2 32424 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
LDual              LCDual

Theoremlcd0vvalN 32425 Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar              LCDual

Theoremlcd0vcl 32426 Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
LCDual

Theoremlcd0vs 32427 A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
Scalar              LCDual

Theoremlcdvs0N 32428 A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
Scalar              LCDual

Theoremlcdvsub 32429 The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
Scalar                     LCDual

Theoremlcdvsubval 32430 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
Scalar              LCDual

Theoremlcdlss 32431* Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual                     LFnl       LKer       LDual

Theoremlcdlss2N 32432 Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
LCDual                            LDual

Theoremlcdlsp 32433 Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
LDual              LCDual

TheoremlcdlkreqN 32434 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
LKer       LCDual

Theoremlcdlkreq2N 32435 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar                     LKer       LCDual

Syntaxcmpd 32436 Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
mapd

Definitiondf-mapd 32437* Extend class notation with a one-to-one onto (mapd1o 32460), order-preserving (mapdord 32450) map, called a projectivity (definition of projectivity in [Baer] p. 40), from subspaces of vector space H to those subspaces of the dual space having functionals with closed kernels. (Contributed by NM, 25-Jan-2015.)
mapd LFnl LKer LKer LKer

Theoremmapdffval 32438* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
mapd LFnl LKer LKer LKer

Theoremmapdfval 32439* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
LFnl       LKer              mapd

Theoremmapdval 32440* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
LFnl       LKer              mapd

Theoremmapdvalc 32441* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
LFnl       LKer              mapd

Theoremmapdval2N 32442* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl       LKer              mapd

Theoremmapdval3N 32443* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl       LKer              mapd

Theoremmapdval4N 32444* Value of projectivity from vector space H to dual space. TODO: 1. This is shorter than others - make it the official def? (but is not as obvious that it is ) 2. The unneeded direction of lcfl8a 32315 has awkward - add another thm with only one direction of it? 3. Swap and ? (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl       LKer              mapd

Theoremmapdval5N 32445* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl       LKer              mapd

Theoremmapdordlem1a 32446* Lemma for mapdord 32450. (Contributed by NM, 27-Jan-2015.)
LSHyp       LFnl       LKer

Theoremmapdordlem1bN 32447* Lemma for mapdord 32450. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)

Theoremmapdordlem1 32448* Lemma for mapdord 32450. (Contributed by NM, 27-Jan-2015.)

Theoremmapdordlem2 32449* Lemma for mapdord 32450. Ordering property of projectivity . TODO: This was proved using some hacked-up older proofs. Maybe simplify; get rid of the hypothesis. (Contributed by NM, 27-Jan-2015.)
mapd                                   LSAtoms       LFnl       LSHyp       LKer

Theoremmapdord 32450 Ordering property of the map defined by df-mapd 32437. Property (b) of [Baer] p. 40. (Contributed by NM, 27-Jan-2015.)
mapd

Theoremmapd11 32451 The map defined by df-mapd 32437 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.)
mapd

TheoremmapddlssN 32452 The mapping of a subspace of vector space H to the dual space is a subspace of the dual space. TODO: Make this obsolete, use mapdcl2 32468 instead. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
mapd                     LDual

Theoremmapdsn 32453* Value of the map defined by df-mapd 32437 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
mapd                            LFnl       LKer

Theoremmapdsn2 32454* Value of the map defined by df-mapd 32437 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
mapd                            LFnl       LKer

Theoremmapdsn3 32455 Value of the map defined by df-mapd 32437 at the span of a singleton. (Contributed by NM, 17-Feb-2015.)
mapd                            LFnl       LKer       LDual

Theoremmapd1dim2lem1N 32456* Value of the map defined by df-mapd 32437 at an atom. (Contributed by NM, 10-Feb-2015.) (New usage is discouraged.)
LSAtoms       LFnl       LKer              mapd

Theoremmapdrvallem2 32457* Lemma for ~? mapdrval . TODO: very long antecendents are dragged through proof in some places - see if it shortens proof to remove unused conjuncts. (Contributed by NM, 2-Feb-2015.)
mapd                     LFnl       LKer       LDual                                                        LSAtoms

Theoremmapdrvallem3 32458* Lemma for ~? mapdrval . (Contributed by NM, 2-Feb-2015.)
mapd                     LFnl       LKer       LDual                                                        LSAtoms

Theoremmapdrval 32459* Given a dual subspace (of functionals with closed kernels), reconstruct the subspace that maps to it. (Contributed by NM, 12-Mar-2015.)
mapd                     LFnl       LKer       LDual

Theoremmapd1o 32460* The map defined by df-mapd 32437 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 32397, mapdrval 32459, lclkrs 32351, lclkr 32345,...) to use ? TODO: maybe get rid of \$d's for vs. ,. propagate to mapdrn 32461 and any others. (Contributed by NM, 12-Mar-2015.)
mapd                     LFnl       LKer       LDual

Theoremmapdrn 32461* Range of the map defined by df-mapd 32437. (Contributed by NM, 12-Mar-2015.)
mapd              LFnl       LKer       LDual

TheoremmapdunirnN 32462* Union of the range of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd              LFnl       LKer

Theoremmapdrn2 32463 Range of the map defined by df-mapd 32437. TODO: this seems to be needed a lot in hdmaprnlem3eN 32673 etc. Would it be better to change df-mapd 32437 theorems to use instead of ? (Contributed by NM, 13-Mar-2015.)
mapd       LCDual

Theoremmapdcnvcl 32464 Closure of the converse of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.)
mapd

Theoremmapdcl 32465 Closure the value of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.)
mapd

Theoremmapdcnvid1N 32466 Converse of the value of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd

Theoremmapdsord 32467 Strong ordering property of themap defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.)
mapd

Theoremmapdcl2 32468 The mapping of a subspace of vector space H is a subspace in the dual space of functionals with closed kernels. (Contributed by NM, 31-Jan-2015.)
mapd                     LCDual

Theoremmapdcnvid2 32469 Value of the converse of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.)
mapd

TheoremmapdcnvordN 32470 Ordering property of the converse of the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd

Theoremmapdcnv11N 32471 The converse of the map defined by df-mapd 32437 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd

Theoremmapdcv 32472 Covering property of the converse of the map defined by df-mapd 32437. (Contributed by NM, 14-Mar-2015.)
mapd                     L        LCDual       L

Theoremmapdincl 32473 Closure of dual subspace intersection for the map defined by df-mapd 32437. (Contributed by NM, 12-Apr-2015.)
mapd              LCDual

Theoremmapdin 32474 Subspace intersection is preserved by the map defined by df-mapd 32437. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)
mapd

Theoremmapdlsmcl 32475 Closure of dual subspace sum for the map defined by df-mapd 32437. (Contributed by NM, 13-Mar-2015.)
mapd              LCDual

Theoremmapdlsm 32476 Subspace sum is preserved by the map defined by df-mapd 32437. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
mapd                            LCDual

Theoremmapd0 32477 Projectivity map of the zero subspace. Part of property (f) in [Baer] p. 40. TODO: does proof need to be this long for this simple fact? (Contributed by NM, 15-Mar-2015.)
mapd                     LCDual

TheoremmapdcnvatN 32478 Atoms are preserved by the map defined by df-mapd 32437. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
mapd              LSAtoms       LCDual       LSAtoms

Theoremmapdat 32479 Atoms are preserved by the map defined by df-mapd 32437. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.)
mapd              LSAtoms       LCDual       LSAtoms

Theoremmapdspex 32480* The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.)
mapd                            LCDual

Theoremmapdn0 32481 Transfer non-zero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.)
mapd                            LCDual

Theoremmapdncol 32482 Transfer non-colinearity from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
mapd                            LCDual

Theoremmapdindp 32483 Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
mapd                            LCDual

Theoremmapdpglem1 32484 Lemma for mapdpg 32518. Baer p. 44, last line: "(F(x-y))* =< (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.)
mapd                                   LCDual

Theoremmapdpglem2 32485* Lemma for mapdpg 32518. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope \$d locally to avoid clashes with later substitutions into .) (Contributed by NM, 15-Mar-2015.)
mapd                                   LCDual

Theoremmapdpglem2a 32486* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual

Theoremmapdpglem3 32487* Lemma for mapdpg 32518. Baer p. 45, line 3: "infer...the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope \$d locally to avoid clashes with later substitutions into .) (Contributed by NM, 18-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem4N 32488* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem5N 32489* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem6 32490* Lemma for mapdpg 32518. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem8 32491* Lemma for mapdpg 32518. Baer p. 45, line 4: "...so that (F(x-y))* =< (Fy)*. This would imply that F(x-y) =< F(y)..." (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem9 32492* Lemma for mapdpg 32518. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem10 32493* Lemma for mapdpg 32518. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem11 32494* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem12 32495* Lemma for mapdpg 32518. TODO: Can some commonality with mapdpglem6 32490 through mapdpglem11 32494 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem13 32496* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem14 32497* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem15 32498* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem16 32499* Lemma for mapdpg 32518. Baer p. 45, line 7: "Likewise we see that z =/= 0." (Contributed by NM, 20-Mar-2015.)
mapd                                   LCDual                                                        Scalar

Theoremmapdpglem17N 32500* Lemma for mapdpg 32518. Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
mapd                                   LCDual                                                        Scalar

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