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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlflcl 32701 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 32702 A linear functional is zero at the zero vector. (lnfn0i 27776 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 32703 Property of a linear functional. (lnfnaddi 27777 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 32704 Property of a linear functional. (lnfnaddi 27777 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 32705 Property of a linear functional. (lnfnmuli 27778 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 32706 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 32707* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 32708 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Scalar              LFnl

Scalar              LFnl

Theoremlfladd0l 32711 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 32712* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 32783, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 32713* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 32783, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 32714 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
Scalar                     LFnl

Theoremlflvsdi1 32715 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 32716 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 32717 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 32718 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 32719 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
Scalar       LFnl

Theoremlflsc0N 32720 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 32721 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 32722 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
LKer

Definitiondf-lkr 32723* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LKer LFnl Scalar

Theoremlkrfval 32724* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar              LFnl       LKer

Theoremlkrval 32725 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr 32726 Membership in the kernel of a functional. (elnlfn 27662 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkrval2 32727* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr2 32728 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Scalar              LFnl       LKer

Theoremlkrcl 32729 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrf0 32730 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkr0f 32731 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
Scalar                     LFnl       LKer

Theoremlkrlss 32732 The kernel of a linear functional is a subspace. (nlelshi 27794 analog.) (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrssv 32733 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlkrsc 32734 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
Scalar                     LFnl       LKer

Theoremlkrscss 32735 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
Scalar                     LFnl       LKer

Theoremeqlkr 32736* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr2 32737* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr3 32738 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremlkrlsp 32739 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 32726) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
Scalar                                   LFnl       LKer

Theoremlkrlsp2 32740 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
LFnl       LKer

Theoremlkrlsp3 32741 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
LFnl       LKer

Theoremlkrshp 32742 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshp3 32743 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshpor 32744 The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
LSHyp       LFnl       LKer

Theoremlkrshp4 32745 A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
LSHyp       LFnl       LKer

Theoremlshpsmreu 32746* Lemma for lshpkrex 32755. Show uniqueness of ring multiplier when a vector is broken down into components, one in a hyperplane and the other outside of it . TODO: do we need the cbvrexv 3006 for to ? (Contributed by NM, 4-Jan-2015.)
LSHyp                                          Scalar

Theoremlshpkrlem1 32747* Lemma for lshpkrex 32755. The value of tentative functional is zero iff its argument belongs to hyperplane . (Contributed by NM, 14-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem2 32748* Lemma for lshpkrex 32755. The value of tentative functional is a scalar. (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem3 32749* Lemma for lshpkrex 32755. Defining property of . (Contributed by NM, 15-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem4 32750* Lemma for lshpkrex 32755. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem5 32751* Lemma for lshpkrex 32755. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem6 32752* Lemma for lshpkrex 32755. Show linearlity of . (Contributed by NM, 17-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrcl 32753* The set defined by hyperplane is a linear functional. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LFnl

Theoremlshpkr 32754* The kernel of functional is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LKer

Theoremlshpkrex 32755* There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
LSHyp       LFnl       LKer

Theoremlshpset2N 32756* The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
Scalar              LSHyp       LFnl       LKer

TheoremislshpkrN 32757* The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be or as in lshpkrex 32755? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar              LSHyp       LFnl       LKer

Theoremlfl1dim 32758* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
Scalar       LFnl       LKer

Theoremlfl1dim2N 32759* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 32758 may be more compatible with lspsn 18303. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Scalar       LFnl       LKer

21.21.8  Opposite rings and dual vector spaces

Syntaxcld 32760 Extend class notation with left dualvector space.
LDual

Definitiondf-ldual 32761* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. The restriction on allows it to be a set; see ofmres 6808. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
LDual LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar

Theoremldualset 32762* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
LFnl       LDual       Scalar                     oppr                     Scalar

Theoremldualvbase 32763 The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
LFnl       LDual

Theoremldualelvbase 32764 Utility theorem for converting a functional to a vector of the dual space in order to use standard vector theorems. (Contributed by NM, 6-Jan-2015.)
LFnl       LDual

Theoremldualfvadd 32765 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvadd 32766 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvaddcl 32767 The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.)
LFnl       LDual

Theoremldualvaddval 32768 The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
Scalar              LFnl       LDual

Theoremldualsca 32769 The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
Scalar       oppr       LDual       Scalar

Theoremldualsbase 32770 Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar              LDual       Scalar

TheoremldualsaddN 32771 Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Scalar              LDual       Scalar

Theoremldualsmul 32772 Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar                     LDual       Scalar

Theoremldualfvs 32773* Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvs 32774 Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvsval 32775 Value of scalar product operation value for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvscl 32776 The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvaddcom 32777 Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
LFnl       LDual

Theoremldualvsass 32778 Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
LFnl       Scalar                     LDual

Theoremldualvsass2 32779 Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
LFnl       Scalar              LDual       Scalar

Theoremldualvsdi1 32780 Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvsdi2 32781 Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar                     LDual

Theoremldualgrplem 32782 Lemma for ldualgrp 32783. (Contributed by NM, 22-Oct-2014.)
LDual                            LFnl       Scalar                     oppr

Theoremldualgrp 32783 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
LDual

Theoremldual0 32784 The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
Scalar              LDual       Scalar

Theoremldual1 32785 The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar              LDual       Scalar

Theoremldualneg 32786 The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar              LDual       Scalar

Theoremldual0v 32787 The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar              LDual

Theoremldual0vcl 32788 The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
LFnl       LDual

Theoremlduallmodlem 32789 Lemma for lduallmod 32790. (Contributed by NM, 22-Oct-2014.)
LDual                            LFnl       Scalar                     oppr

Theoremlduallmod 32790 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
LDual

Theoremlduallvec 32791 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 17929; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
LDual

Theoremldualvsub 32792 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
Scalar                     LFnl       LDual

Theoremldualvsubcl 32793 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
LFnl       LDual

Theoremldualvsubval 32794 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 32792? (Requires to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
Scalar              LFnl       LDual

Theoremldualssvscl 32795 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
Scalar              LDual

Theoremldualssvsubcl 32796 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
LDual

Theoremldual0vs 32797 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
LFnl       Scalar              LDual

Theoremlkr0f2 32798 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
LFnl       LKer       LDual

Theoremlduallkr3 32799 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
LSHyp       LFnl       LKer       LDual

TheoremlkrpssN 32800 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
LFnl       LKer       LDual

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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