P. 347 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 34601-34700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	lcfrlem26 34601*	Lemma for lcfr 34618. Special case of lcfrlem36 34611 when is zero. (Contributed by NM, 11-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem27 34602*	Lemma for lcfr 34618. Special case of lcfrlem37 34612 when is zero. (Contributed by NM, 11-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual LFnl

Theorem	lcfrlem28 34603*	Lemma for lcfr 34618. TODO: This can be a hypothesis since the zero version of needs it. (Contributed by NM, 9-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem29 34604*	Lemma for lcfr 34618. (Contributed by NM, 9-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem30 34605*	Lemma for lcfr 34618. (Contributed by NM, 6-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual LFnl

Theorem	lcfrlem31 34606*	Lemma for lcfr 34618. (Contributed by NM, 10-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem32 34607*	Lemma for lcfr 34618. (Contributed by NM, 10-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem33 34608*	Lemma for lcfr 34618. (Contributed by NM, 10-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem34 34609*	Lemma for lcfr 34618. (Contributed by NM, 10-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem35 34610*	Lemma for lcfr 34618. (Contributed by NM, 2-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem36 34611*	Lemma for lcfr 34618. (Contributed by NM, 6-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual

Theorem	lcfrlem37 34612*	Lemma for lcfr 34618. (Contributed by NM, 8-Mar-2015.)
LSAtoms $/\$ $\$ $\$ Scalar $\$ LKer LDual LFnl

Theorem	lcfrlem38 34613*	Lemma for lcfr 34618. Combine lcfrlem27 34602 and lcfrlem37 34612. (Contributed by NM, 11-Mar-2015.)
LFnl LKer LDual LFnl $/\$ Scalar $\$

Theorem	lcfrlem39 34614*	Lemma for lcfr 34618. Eliminate . (Contributed by NM, 11-Mar-2015.)
LFnl LKer LDual LFnl $/\$

Theorem	lcfrlem40 34615*	Lemma for lcfr 34618. Eliminate and . (Contributed by NM, 11-Mar-2015.)
LFnl LKer LDual LFnl $/\$

Theorem	lcfrlem41 34616*	Lemma for lcfr 34618. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
LFnl LKer LDual LFnl $/\$

Theorem	lcfrlem42 34617*	Lemma for lcfr 34618. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
LFnl LKer LDual LFnl $/\$

Theorem	lcfr 34618*	Reconstruction of a subspace from a dual subspace of functionals with closed kernels. Our proof was suggested by Mario Carneiro, 20-Feb-2015. (Contributed by NM, 5-Mar-2015.)
LFnl LKer LDual $/\$

Syntax	clcd 34619	Extend class notation with vector space of functionals with closed kernels.
LCDual

Definition	df-lcdual 34620*	Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 34682. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 34658 using LCDual. (Contributed by NM, 13-Mar-2015.)
LCDual LDual ↾_s LFnl LKer LKer

Theorem	lcdfval 34621*	Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual LDual ↾_s LFnl LKer LKer

Theorem	lcdval 34622*	Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual LFnl LKer LDual $/\$ ↾_s

Theorem	lcdval2 34623*	Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual LFnl LKer LDual $/\$ ↾_s

Theorem	lcdlvec 34624	The dual vector space of functionals with closed kernels is a left vector space. (Contributed by NM, 14-Mar-2015.)
LCDual $/\$

Theorem	lcdlmod 34625	The dual vector space of functionals with closed kernels is a left module. (Contributed by NM, 13-Mar-2015.)
LCDual $/\$

Theorem	lcdvbase 34626*	Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual LFnl LKer $/\$

Theorem	lcdvbasess 34627	The vector base set of the closed kernel dual space is a set of functionals. (Contributed by NM, 15-Mar-2015.)
LCDual LFnl $/\$

Theorem	lcdvbaselfl 34628	A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.)
LCDual LFnl $/\$

Theorem	lcdvbasecl 34629	Closure of the value of a vector (functional) in the closed kernel dual space. (Contributed by NM, 28-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdvadd 34630	Vector addition for the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
LDual LCDual $/\$

Theorem	lcdvaddval 34631	The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
Scalar LCDual $/\$

Theorem	lcdsca 34632	The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Scalar opp_r LCDual Scalar $/\$

Theorem	lcdsbase 34633	Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcdsadd 34634	Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcdsmul 34635	Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcdvs 34636	Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
LDual LCDual $/\$

Theorem	lcdvsval 34637	Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdvscl 34638	The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdlssvscl 34639	Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdvsass 34640	Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcd0 34641	The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcd1 34642	The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcdneg 34643	The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
Scalar LCDual Scalar $/\$

Theorem	lcd0v 34644	The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcd0v2 34645	The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
LDual LCDual $/\$

Theorem	lcd0vvalN 34646	Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar LCDual $/\$

Theorem	lcd0vcl 34647	Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
LCDual $/\$

Theorem	lcd0vs 34648	A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdvs0N 34649	A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
Scalar LCDual $/\$

Theorem	lcdvsub 34650	The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
Scalar LCDual $/\$

Theorem	lcdvsubval 34651	The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
Scalar LCDual $/\$

Theorem	lcdlss 34652*	Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
LCDual LFnl LKer LDual $/\$

Theorem	lcdlss2N 34653	Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
LCDual LDual $/\$

Theorem	lcdlsp 34654	Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
LDual LCDual $/\$

Theorem	lcdlkreqN 34655	Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
LKer LCDual $/\$

Theorem	lcdlkreq2N 34656	Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar LKer LCDual $/\$ $\$

Syntax	cmpd 34657	Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
mapd

Definition	df-mapd 34658*	Extend class notation with a one-to-one onto (mapd1o 34681), order-preserving (mapdord 34671) 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

Theorem	mapdffval 34659*	Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
mapd LFnl LKer LKer $/\$ LKer

Theorem	mapdfval 34660*	Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
LFnl LKer mapd $/\$ $/\$

Theorem	mapdval 34661*	Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
LFnl LKer mapd $/\$ $/\$

Theorem	mapdvalc 34662*	Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
LFnl LKer mapd $/\$

Theorem	mapdval2N 34663*	Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl LKer mapd $/\$

Theorem	mapdval3N 34664*	Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl LKer mapd $/\$

Theorem	mapdval4N 34665*	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 34536 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 $/\$

Theorem	mapdval5N 34666*	Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
LFnl LKer mapd $/\$

Theorem	mapdordlem1a 34667*	Lemma for mapdord 34671. (Contributed by NM, 27-Jan-2015.)
LSHyp LFnl LKer $/\$ $/\$

Theorem	mapdordlem1bN 34668*	Lemma for mapdord 34671. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
$/\$

Theorem	mapdordlem1 34669*	Lemma for mapdord 34671. (Contributed by NM, 27-Jan-2015.)
$/\$

Theorem	mapdordlem2 34670*	Lemma for mapdord 34671. 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

Theorem	mapdord 34671	Ordering property of the map defined by df-mapd 34658. Property (b) of [Baer] p. 40. (Contributed by NM, 27-Jan-2015.)
mapd $/\$

Theorem	mapd11 34672	The map defined by df-mapd 34658 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.)
mapd $/\$

Theorem	mapddlssN 34673	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 34689 instead. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
mapd LDual $/\$

Theorem	mapdsn 34674*	Value of the map defined by df-mapd 34658 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
mapd LFnl LKer $/\$

Theorem	mapdsn2 34675*	Value of the map defined by df-mapd 34658 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
mapd LFnl LKer $/\$

Theorem	mapdsn3 34676	Value of the map defined by df-mapd 34658 at the span of a singleton. (Contributed by NM, 17-Feb-2015.)
mapd LFnl LKer LDual $/\$

Theorem	mapd1dim2lem1N 34677*	Value of the map defined by df-mapd 34658 at an atom. (Contributed by NM, 10-Feb-2015.) (New usage is discouraged.)
LSAtoms LFnl LKer mapd $/\$

Theorem	mapdrvallem2 34678*	Lemma for mapdrval 34680. TODO: very long antecedents 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

Theorem	mapdrvallem3 34679*	Lemma for mapdrval 34680. (Contributed by NM, 2-Feb-2015.)
mapd LFnl LKer LDual $/\$ LSAtoms

Theorem	mapdrval 34680*	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 $/\$

Theorem	mapd1o 34681*	The map defined by df-mapd 34658 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 34618, mapdrval 34680, lclkrs 34572, lclkr 34566,...) to use ? TODO: maybe get rid of $d's for vs. ,. propagate to mapdrn 34682 and any others. (Contributed by NM, 12-Mar-2015.)
mapd LFnl LKer LDual $/\$

Theorem	mapdrn 34682*	Range of the map defined by df-mapd 34658. (Contributed by NM, 12-Mar-2015.)
mapd LFnl LKer LDual $/\$

Theorem	mapdunirnN 34683*	Union of the range of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd LFnl LKer $/\$

Theorem	mapdrn2 34684	Range of the map defined by df-mapd 34658. TODO: this seems to be needed a lot in hdmaprnlem3eN 34894 etc. Would it be better to change df-mapd 34658 theorems to use instead of ? (Contributed by NM, 13-Mar-2015.)
mapd LCDual $/\$

Theorem	mapdcnvcl 34685	Closure of the converse of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.)
mapd $/\$

Theorem	mapdcl 34686	Closure the value of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.)
mapd $/\$

Theorem	mapdcnvid1N 34687	Converse of the value of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd $/\$

Theorem	mapdsord 34688	Strong ordering property of themap defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.)
mapd $/\$

Theorem	mapdcl2 34689	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 $/\$

Theorem	mapdcnvid2 34690	Value of the converse of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.)
mapd $/\$

Theorem	mapdcnvordN 34691	Ordering property of the converse of the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd $/\$

Theorem	mapdcnv11N 34692	The converse of the map defined by df-mapd 34658 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
mapd $/\$

Theorem	mapdcv 34693	Covering property of the converse of the map defined by df-mapd 34658. (Contributed by NM, 14-Mar-2015.)
mapd _L LCDual _L $/\$

Theorem	mapdincl 34694	Closure of dual subspace intersection for the map defined by df-mapd 34658. (Contributed by NM, 12-Apr-2015.)
mapd LCDual $/\$

Theorem	mapdin 34695	Subspace intersection is preserved by the map defined by df-mapd 34658. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)
mapd $/\$

Theorem	mapdlsmcl 34696	Closure of dual subspace sum for the map defined by df-mapd 34658. (Contributed by NM, 13-Mar-2015.)
mapd LCDual $/\$

Theorem	mapdlsm 34697	Subspace sum is preserved by the map defined by df-mapd 34658. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
mapd LCDual $/\$

Theorem	mapd0 34698	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 $/\$

Theorem	mapdcnvatN 34699	Atoms are preserved by the map defined by df-mapd 34658. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
mapd LSAtoms LCDual LSAtoms $/\$

Theorem	mapdat 34700	Atoms are preserved by the map defined by df-mapd 34658. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.)
mapd LSAtoms LCDual LSAtoms $/\$

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