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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dochexmidlem3 35101 | Lemma for dochexmid 35107. Use atom exchange lsatexch1 32683 to swap and . (Contributed by NM, 14-Jan-2015.) |
LSAtoms | ||
Theorem | dochexmidlem4 35102 | Lemma for dochexmid 35107. (Contributed by NM, 15-Jan-2015.) |
LSAtoms | ||
Theorem | dochexmidlem5 35103 | Lemma for dochexmid 35107. (Contributed by NM, 15-Jan-2015.) |
LSAtoms | ||
Theorem | dochexmidlem6 35104 | Lemma for dochexmid 35107. (Contributed by NM, 15-Jan-2015.) |
LSAtoms | ||
Theorem | dochexmidlem7 35105 | Lemma for dochexmid 35107. Contradict dochexmidlem6 35104. (Contributed by NM, 15-Jan-2015.) |
LSAtoms | ||
Theorem | dochexmidlem8 35106 | Lemma for dochexmid 35107. The contradiction of dochexmidlem6 35104 and dochexmidlem7 35105 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 | ||
Theorem | dochexmid 35107 | Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 35016. 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 33614 analog.) (Contributed by NM, 15-Jan-2015.) |
Theorem | dochsnkrlem1 35108 | Lemma for dochsnkr 35111. (Contributed by NM, 1-Jan-2015.) |
LFnl LKer | ||
Theorem | dochsnkrlem2 35109 | Lemma for dochsnkr 35111. (Contributed by NM, 1-Jan-2015.) |
LFnl LKer LSAtoms | ||
Theorem | dochsnkrlem3 35110 | Lemma for dochsnkr 35111. (Contributed by NM, 2-Jan-2015.) |
LFnl LKer | ||
Theorem | dochsnkr 35111 | 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 | ||
Theorem | dochsnkr2 35112* | Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 32754. (Contributed by NM, 27-Oct-2014.) |
LKer Scalar | ||
Theorem | dochsnkr2cl 35113* | The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.) |
LKer Scalar | ||
Theorem | dochflcl 35114* | Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 32753. (Contributed by NM, 27-Oct-2014.) |
LFnl Scalar | ||
Theorem | dochfl1 35115* | The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.) |
Scalar | ||
Theorem | dochfln0 35116 | 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 | ||
Theorem | dochkr1 35117* | 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 32707. (Contributed by NM, 2-Jan-2015.) |
Scalar LFnl LKer | ||
Theorem | dochkr1OLDN 35118* | 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 32707. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.) |
Scalar LFnl LKer | ||
Syntax | clpoN 35119 | Extend class notation with all polarities of a left module or left vector space. |
LPol | ||
Definition | df-lpolN 35120* | 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 | ||
Theorem | lpolsetN 35121* | 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 | ||
Theorem | islpolN 35122* | The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.) |
LSAtoms LSHyp LPol | ||
Theorem | islpoldN 35123* | Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LSAtoms LSHyp LPol | ||
Theorem | lpolfN 35124 | Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LPol | ||
Theorem | lpolvN 35125 | The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LPol | ||
Theorem | lpolconN 35126 | Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LPol | ||
Theorem | lpolsatN 35127 | The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LSAtoms LSHyp LPol | ||
Theorem | lpolpolsatN 35128 | Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
LSAtoms LPol | ||
Theorem | dochpolN 35129 | The subspace orthocomplement for the vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.) |
LPol | ||
Theorem | lcfl1lem 35130* | Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.) |
Theorem | lcfl1 35131* | Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.) |
Theorem | lcfl2 35132* | Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) |
LFnl LKer | ||
Theorem | lcfl3 35133* | Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) |
LSAtoms LFnl LKer | ||
Theorem | lcfl4N 35134* | Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.) |
LSHyp LFnl LKer | ||
Theorem | lcfl5 35135* | Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) |
LFnl LKer | ||
Theorem | lcfl5a 35136 | Property of a functional with a closed kernel. TODO: Make lcfl5 35135 etc. obsolete and rewrite w/out hypothesis? (Contributed by NM, 29-Jan-2015.) |
LFnl LKer | ||
Theorem | lcfl6lem 35137* | Lemma for lcfl6 35139. A functional (whose kernel is closed by dochsnkr 35111) 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 | ||
Theorem | lcfl7lem 35138* | Lemma for lcfl7N 35140. If two functionals and are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.) |
Scalar LFnl LKer | ||
Theorem | lcfl6 35139* | Property of a functional with a closed kernel. Note that means the functional is zero by lkr0f 32731. (Contributed by NM, 3-Jan-2015.) |
Scalar LFnl LKer | ||
Theorem | lcfl7N 35140* | 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 32731. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.) |
Scalar LFnl LKer | ||
Theorem | lcfl8 35141* | Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.) |
LFnl LKer | ||
Theorem | lcfl8a 35142* | Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.) |
LFnl LKer | ||
Theorem | lcfl8b 35143* | Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.) |
LFnl LKer LDual | ||
Theorem | lcfl9a 35144 | Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.) |
LFnl LKer | ||
Theorem | lclkrlem1 35145* | The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.) |
LFnl LKer LDual Scalar | ||
Theorem | lclkrlem2a 35146 | Lemma for lclkr 35172. Use lshpat 32693 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.) |
LSAtoms | ||
Theorem | lclkrlem2b 35147 | Lemma for lclkr 35172. (Contributed by NM, 17-Jan-2015.) |
LSAtoms | ||
Theorem | lclkrlem2c 35148 | Lemma for lclkr 35172. (Contributed by NM, 16-Jan-2015.) |
LSAtoms LSHyp | ||
Theorem | lclkrlem2d 35149 | Lemma for lclkr 35172. (Contributed by NM, 16-Jan-2015.) |
LSAtoms | ||
Theorem | lclkrlem2e 35150 | Lemma for lclkr 35172. 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 | ||
Theorem | lclkrlem2f 35151 | Lemma for lclkr 35172. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2g 35152 | Lemma for lclkr 35172. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2h 35153 | Lemma for lclkr 35172. Eliminate the hypothesis. (Contributed by NM, 16-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2i 35154 | Lemma for lclkr 35172. Eliminate the hypothesis. (Contributed by NM, 17-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2j 35155 | Lemma for lclkr 35172. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2k 35156 | Lemma for lclkr 35172. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2l 35157 | Lemma for lclkr 35172. Eliminate the , hypotheses. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LSHyp LKer LDual | ||
Theorem | lclkrlem2m 35158 | Lemma for lclkr 35172. 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 | ||
Theorem | lclkrlem2n 35159 | Lemma for lclkr 35172. (Contributed by NM, 12-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2o 35160 | Lemma for lclkr 35172. 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 | ||
Theorem | lclkrlem2p 35161 | Lemma for lclkr 35172. When is zero, and must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2q 35162 | Lemma for lclkr 35172. The sum has a closed kernel when is nonzero. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2r 35163 | Lemma for lclkr 35172. When is zero, i.e. when and are colinear, the intersection of the kernels of and equal the kernel of , so the kernels of and the sum are comparable. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2s 35164 | Lemma for lclkr 35172. Thus, the sum has a closed kernel when is zero. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2t 35165 | Lemma for lclkr 35172. We eliminate all hypotheses with here. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2u 35166 | Lemma for lclkr 35172. lclkrlem2t 35165 with and swapped. (Contributed by NM, 18-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2v 35167 | Lemma for lclkr 35172. When the hypotheses of lclkrlem2u 35166 and lclkrlem2u 35166 are negated, the functional sum must be zero, so the kernel is the vector space. We make use of the law of excluded middle, dochexmid 35107, which requires the orthomodular law dihoml4 35016 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2w 35168 | Lemma for lclkr 35172. This is the same as lclkrlem2u 35166 and lclkrlem2u 35166 with the inequality hypotheses negated. When the sum of two functionals is zero at each generating vector, the kernel is the vector space and therefore closed. (Contributed by NM, 16-Jan-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lclkrlem2x 35169 | Lemma for lclkr 35172. Eliminate by cases the hypotheses of lclkrlem2u 35166, lclkrlem2u 35166 and lclkrlem2w 35168. (Contributed by NM, 18-Jan-2015.) |
LKer LFnl LDual | ||
Theorem | lclkrlem2y 35170 | Lemma for lclkr 35172. Restate the hypotheses for and to say their kernels are closed, in order to eliminate the generating vectors and . (Contributed by NM, 18-Jan-2015.) |
LKer LFnl LDual | ||
Theorem | lclkrlem2 35171* | The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 35146 through lclkrlem2y 35170 are used for the proof. Here we express lclkrlem2y 35170 in terms of membership in the set of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.) |
LFnl LKer LDual | ||
Theorem | lclkr 35172* | The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The f_{M} that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.) |
LFnl LKer LDual | ||
Theorem | lcfls1lem 35173* | Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) |
Theorem | lcfls1N 35174* | Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.) |
Theorem | lcfls1c 35175* | Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.) |
Theorem | lclkrslem1 35176* | The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace is closed under scalar product. (Contributed by NM, 27-Jan-2015.) |
LFnl LKer LDual Scalar | ||
Theorem | lclkrslem2 35177* | The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace is closed under scalar product. (Contributed by NM, 28-Jan-2015.) |
LFnl LKer LDual Scalar | ||
Theorem | lclkrs 35178* | The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 35172 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 35172 a special case of this? (Contributed by NM, 29-Jan-2015.) |
LFnl LKer LDual | ||
Theorem | lclkrs2 35179* | The set of functionals with closed kernels and majorizing the orthocomplement of a given subspace is a subspace of the dual space containing functionals with closed kernels. Note that is the value given by mapdval 35267. (Contributed by NM, 12-Mar-2015.) |
LFnl LKer LDual | ||
Theorem | lcfrvalsnN 35180* | Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.) |
LFnl LKer LDual | ||
Theorem | lcfrlem1 35181 | Lemma for lcfr 35224. Note that is z in Mario's notes. (Contributed by NM, 27-Feb-2015.) |
Scalar LFnl LDual | ||
Theorem | lcfrlem2 35182 | Lemma for lcfr 35224. (Contributed by NM, 27-Feb-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lcfrlem3 35183 | Lemma for lcfr 35224. (Contributed by NM, 27-Feb-2015.) |
Scalar LFnl LDual LKer | ||
Theorem | lcfrlem4 35184* | Lemma for lcfr 35224. (Contributed by NM, 10-Mar-2015.) |
LKer LDual | ||
Theorem | lcfrlem5 35185* | Lemma for lcfr 35224. The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace is closed under scalar product. TODO: share hypotheses with others. Use more consistent variable names here or elsewhere when possible. (Contributed by NM, 5-Mar-2015.) |
LFnl LKer LDual Scalar | ||
Theorem | lcfrlem6 35186* | Lemma for lcfr 35224. Closure of vector sum with colinear vectors. TODO: Move down definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.) |
LKer LDual | ||
Theorem | lcfrlem7 35187* | Lemma for lcfr 35224. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.) |
LKer LDual |