P. 351 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 35001-35100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dochexmidlem5 35001	Lemma for dochexmid 35005. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem6 35002	Lemma for dochexmid 35005. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem7 35003	Lemma for dochexmid 35005. Contradict dochexmidlem6 35002. (Contributed by NM, 15-Jan-2015.)
LSAtoms $/\$

Theorem	dochexmidlem8 35004	Lemma for dochexmid 35005. The contradiction of dochexmidlem6 35002 and dochexmidlem7 35003 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 35005	Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 34914. 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 33512 analog.) (Contributed by NM, 15-Jan-2015.)
$/\$

Theorem	dochsnkrlem1 35006	Lemma for dochsnkr 35009. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkrlem2 35007	Lemma for dochsnkr 35009. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\$ LSAtoms

Theorem	dochsnkrlem3 35008	Lemma for dochsnkr 35009. (Contributed by NM, 2-Jan-2015.)
LFnl LKer $/\$ $\$

Theorem	dochsnkr 35009	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 35010*	Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 32652. (Contributed by NM, 27-Oct-2014.)
LKer Scalar $/\$ $\$

Theorem	dochsnkr2cl 35011*	The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer Scalar $/\$ $\$ $\$

Theorem	dochflcl 35012*	Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 32651. (Contributed by NM, 27-Oct-2014.)
LFnl Scalar $/\$ $\$

Theorem	dochfl1 35013*	The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar $/\$ $\$

Theorem	dochfln0 35014	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 35015*	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 32605. (Contributed by NM, 2-Jan-2015.)
Scalar LFnl LKer $/\$ $\$

Theorem	dochkr1OLDN 35016*	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 32605. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar LFnl LKer $/\$

21.21.14 Construction of involution and inner product from a Hilbert lattice

Syntax	clpoN 35017	Extend class notation with all polarities of a left module or left vector space.
LPol

Definition	df-lpolN 35018*	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 35019*	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 35020*	The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	islpoldN 35021*	Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lpolfN 35022	Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolvN 35023	The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolconN 35024	Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

Theorem	lpolsatN 35025	The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LSHyp LPol

Theorem	lpolpolsatN 35026	Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms LPol

Theorem	dochpolN 35027	The subspace orthocomplement for the vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
LPol $/\$

Theorem	lcfl1lem 35028*	Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
$/\$

Theorem	lcfl1 35029*	Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)


Theorem	lcfl2 35030*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$ $\/$

Theorem	lcfl3 35031*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LSAtoms LFnl LKer $/\$ $\/$

Theorem	lcfl4N 35032*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
LSHyp LFnl LKer $/\$ $\/$

Theorem	lcfl5 35033*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl5a 35034	Property of a functional with a closed kernel. TODO: Make lcfl5 35033 etc. obsolete and rewrite w/out hypothesis? (Contributed by NM, 29-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl6lem 35035*	Lemma for lcfl6 35037. A functional (whose kernel is closed by dochsnkr 35009) 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 35036*	Lemma for lcfl7N 35038. If two functionals and are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
Scalar LFnl LKer $/\$ $\$ $\$

Theorem	lcfl6 35037*	Property of a functional with a closed kernel. Note that means the functional is zero by lkr0f 32629. (Contributed by NM, 3-Jan-2015.)
Scalar LFnl LKer $/\$ $\/$ $\$

Theorem	lcfl7N 35038*	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 32629. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
Scalar LFnl LKer $/\$ $\/$ $\$

Theorem	lcfl8 35039*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl8a 35040*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
LFnl LKer $/\$

Theorem	lcfl8b 35041*	Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
LFnl LKer LDual $/\$ $\$ $\$

Theorem	lcfl9a 35042	Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
LFnl LKer $/\$

Theorem	lclkrlem1 35043*	The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
LFnl LKer LDual Scalar $/\$

Theorem	lclkrlem2a 35044	Lemma for lclkr 35070. Use lshpat 32591 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 35045	Lemma for lclkr 35070. (Contributed by NM, 17-Jan-2015.)
LSAtoms $/\$ $\$ $\$ $\$ $\/$

Theorem	lclkrlem2c 35046	Lemma for lclkr 35070. (Contributed by NM, 16-Jan-2015.)
LSAtoms $/\$ $\$ $\$ $\$ $\/$ LSHyp

Theorem	lclkrlem2d 35047	Lemma for lclkr 35070. (Contributed by NM, 16-Jan-2015.)
LSAtoms $/\$ $\$ $\$ $\$ $\/$

Theorem	lclkrlem2e 35048	Lemma for lclkr 35070. 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 35049	Lemma for lclkr 35070. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$ $\$ $\$

Theorem	lclkrlem2g 35050	Lemma for lclkr 35070. 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 35051	Lemma for lclkr 35070. Eliminate the hypothesis. (Contributed by NM, 16-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$ $\$ $\$

Theorem	lclkrlem2i 35052	Lemma for lclkr 35070. Eliminate the hypothesis. (Contributed by NM, 17-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$ $\$ $\$

Theorem	lclkrlem2j 35053	Lemma for lclkr 35070. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$

Theorem	lclkrlem2k 35054	Lemma for lclkr 35070. Kernel closure when is zero. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$

Theorem	lclkrlem2l 35055	Lemma for lclkr 35070. Eliminate the , hypotheses. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LSHyp LKer LDual $/\$ $\$ $\/$

Theorem	lclkrlem2m 35056	Lemma for lclkr 35070. 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 35057	Lemma for lclkr 35070. (Contributed by NM, 12-Jan-2015.)
Scalar LFnl LDual LKer

Theorem	lclkrlem2o 35058	Lemma for lclkr 35070. 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 35059	Lemma for lclkr 35070. When is zero, and must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2q 35060	Lemma for lclkr 35070. The sum has a closed kernel when is nonzero. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2r 35061	Lemma for lclkr 35070. 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 35062	Lemma for lclkr 35070. Thus, the sum has a closed kernel when is zero. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2t 35063	Lemma for lclkr 35070. We eliminate all hypotheses with here. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2u 35064	Lemma for lclkr 35070. lclkrlem2t 35063 with and swapped. (Contributed by NM, 18-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2v 35065	Lemma for lclkr 35070. When the hypotheses of lclkrlem2u 35064 and lclkrlem2u 35064 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 35005, which requires the orthomodular law dihoml4 34914 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)
Scalar LFnl LDual LKer $/\$

Theorem	lclkrlem2w 35066	Lemma for lclkr 35070. This is the same as lclkrlem2u 35064 and lclkrlem2u 35064 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 35067	Lemma for lclkr 35070. Eliminate by cases the hypotheses of lclkrlem2u 35064, lclkrlem2u 35064 and lclkrlem2w 35066. (Contributed by NM, 18-Jan-2015.)
LKer LFnl LDual $/\$

Theorem	lclkrlem2y 35068	Lemma for lclkr 35070. 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 35069*	The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 35044 through lclkrlem2y 35068 are used for the proof. Here we express lclkrlem2y 35068 in terms of membership in the set of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
LFnl LKer LDual $/\$

Theorem	lclkr 35070*	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 35071*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
$/\$ $/\$ $/\$

Theorem	lcfls1N 35072*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lcfls1c 35073*	Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
$/\$ $/\$

Theorem	lclkrslem1 35074*	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 35075*	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 35076*	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 35070 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 35070 a special case of this? (Contributed by NM, 29-Jan-2015.)
LFnl LKer LDual $/\$ $/\$

Theorem	lclkrs2 35077*	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 35165. (Contributed by NM, 12-Mar-2015.)
LFnl LKer LDual $/\$ $/\$ $/\$

Theorem	lcfrvalsnN 35078*	Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.)
LFnl LKer LDual $/\$

Theorem	lcfrlem1 35079	Lemma for lcfr 35122. Note that is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
Scalar LFnl LDual

Theorem	lcfrlem2 35080	Lemma for lcfr 35122. (Contributed by NM, 27-Feb-2015.)
Scalar LFnl LDual LKer

Theorem	lcfrlem3 35081	Lemma for lcfr 35122. (Contributed by NM, 27-Feb-2015.)
Scalar LFnl LDual LKer

Theorem	lcfrlem4 35082*	Lemma for lcfr 35122. (Contributed by NM, 10-Mar-2015.)
LKer LDual $/\$

Theorem	lcfrlem5 35083*	Lemma for lcfr 35122. 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 35084*	Lemma for lcfr 35122. 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 35085*	Lemma for lcfr 35122. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
LKer LDual $/\$

Theorem	lcfrlem8 35086*	Lemma for lcf1o 35088 and lcfr 35122. (Contributed by NM, 21-Feb-2015.)
Scalar LFnl LKer LDual $\$ $/\$ $\$

Theorem	lcfrlem9 35087*	Lemma for lcf1o 35088. (This part has undesirable $d's on and that we remove in lcf1o 35088.) TODO: ugly proof; maybe have better subtheorems or abbreviate some expansions with ? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)