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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorema12studyALT 28401 Alternate proof of a12study 28400, also without using ax12o 1877. (Contributed by NM, 17-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. z  z  =  y  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )   &    |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y )   =>    |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theorema12study2 28402* Reprove ax12o 1877 using dvelimfALT2 1737, showing that ax12o 1877 can be replaced by dveeq2 1882 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1604. (Contributed by Andrew Salmon, 21-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  z  ->  ( w  =  z  ->  A. x  w  =  z )
 )   &    |-  ( -.  A. x  x  =  y  ->  ( w  =  y  ->  A. x  w  =  y ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theorema12study3 28403 Rederivation of axiom ax12o 1877 from two other formulas, without using ax12o 1877. See equvini 1930 and equveli 1931 for the proofs of the hypotheses (using ax12o 1877). Although the second hypothesis (when expanded to primitives) is longer than ax12o 1877, an open problem is whether it can be derived without ax12o 1877 or from a simpler axiom.

Note also that the proof depends on ax-11o 2084, whose proof ax11o 1938 depends on ax12o 1877, meaning that we would have to replace ax-11 1716 with ax-11o 2084 in an axiomatization that uses the hypotheses in place of ax12o 1877. Whether this can be avoided is an open problem. (Contributed by NM, 1-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )   &    |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )   =>    |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theorema12study10 28404* Experiment to study ax12o 1877. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. z ( z  =  x  /\  x  =  y )  ->  A. z
 ( z  =  x 
 ->  x  =  y
 ) )
 
Theorema12study10n 28405* Experiment to study ax12o 1877. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. z ( z  =  x  /\  -.  x  =  y )  ->  A. z
 ( z  =  x 
 ->  -.  x  =  y ) )
 
Theorema12study11 28406* Experiment to study ax12o 1877. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  z  =  x  ->  ( x  =  y 
 ->  A. z  x  =  y ) )   =>    |-  ( E. z  x  =  y  ->  A. z ( z  =  x  ->  x  =  y ) )
 
Theorema12study11n 28407* Experiment to study ax12o 1877. (Contributed by NM, 16-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  z  =  x  ->  ( -.  x  =  y  ->  A. z  -.  x  =  y )
 )   =>    |-  ( E. z  -.  x  =  y  ->  A. z ( z  =  x  ->  -.  x  =  y ) )
 
Theoremax9lem1 28408* Lemma for ax9 1891. Similar to equcomi 1647, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   =>    |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax9lem2 28409* Lemma for ax9 1891. Similar to equequ2 1650, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  z   &    |-  -.  A. w  -.  w  =  x   =>    |-  ( x  =  y  ->  ( z  =  x  <->  z  =  y
 ) )
 
Theoremax9lem3 28410* Lemma for ax9 1891. Similar to sp 1717, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   =>    |-  ( A. x ph  ->  ph )
 
Theoremax9lem4 28411* Lemma for ax9 1891. Similar to ax9o 1892, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  y   =>    |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Theoremax9lem5 28412* Lemma for ax9 1891. Similar to spim 1918 with distinct variables, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  y   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremax9lem6 28413* Lemma for ax9 1891. Helps reduce the number of hypotheses. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  y   &    |-  -.  A. y  -.  y  =  z   =>    |-  -.  A. x  -.  x  =  z
 
Theoremax9lem7 28414* Lemma for ax9 1891. Similar to hba1 1721, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   =>    |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremax9lem8 28415* Lemma for ax9 1891. Similar to hbn 1722, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremax9lem9 28416* Lemma for ax9 1891. Similar to hbimd 1723, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremax9lem10 28417* Lemma for ax9 1891. Similar to hban 1738, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  (
 ( ph  /\  ps )  ->  A. x ( ph  /\ 
 ps ) )
 
Theoremax9lem11 28418* Lemma for ax9 1891. Similar to exlimih 1731, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremax9lem12 28419* Lemma for ax9 1891. Similar to spime 1919 with distinct variables, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  y   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ps )
 
Theoremax9lem13 28420* Lemma for ax9 1891. Similar to cbv3 1925 with distinct variables, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  y   &    |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremax9lem14 28421* Change bound variable without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 22-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. z  -.  z  =  x   &    |-  -.  A. x  -.  x  =  z   &    |-  -.  A. x  -.  x  =  v   &    |-  -.  A. z  -.  z  =  v   &    |-  -.  A. v  -.  v  =  z   &    |-  -.  A. v  -.  v  =  y   =>    |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremax9lem15 28422* Change free variable without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   &    |-  -.  A. x  -.  x  =  z   &    |-  -.  A. x  -.  x  =  w   =>    |-  ( A. x  x  =  y  ->  A. x  x  =  z )
 
Theoremax9lem16 28423* Lemma for ax9 1891. Similar to ax-10 2083 but with distinct variables, without using sp 1717, ax9 1891, or ax-10 2083. We used ax9lem6 28413 to eliminate 5 hypotheses that would otherwise be needed. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. v  -.  v  =  x   &    |-  -.  A. v  -.  v  =  y   &    |-  -.  A. w  -.  w  =  x   &    |-  -.  A. w  -.  w  =  z   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. x  -.  x  =  z   &    |-  -.  A. y  -.  y  =  v   &    |-  -.  A. y  -.  y  =  w   &    |-  -.  A. z  -.  z  =  v   &    |-  -.  A. z  -.  z  =  w   =>    |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremax9lem17 28424* Lemma for ax9 1891. Similar to dvelim 1962 with first hypothesis replaced by distinct variable condition, without using sp 1717, ax9 1891, or ax-10 2083. We used ax9lem6 28413 to eliminate 3 hypotheses that would otherwise be needed. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. u  -.  u  =  v   &    |-  -.  A. u  -.  u  =  w   &    |-  -.  A. v  -.  v  =  x   &    |-  -.  A. v  -.  v  =  z   &    |-  -.  A. w  -.  w  =  x   &    |-  -.  A. w  -.  w  =  z   &    |-  -.  A. x  -.  x  =  u   &    |-  -.  A. x  -.  x  =  w   &    |-  -.  A. z  -.  z  =  u   &    |-  -.  A. z  -.  z  =  w   &    |-  -.  A. z  -.  z  =  y   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax9lem18 28425* Lemma for ax9 1891. Similar to dveeq2 1882, without using sp 1717, ax9 1891, or ax-10 2083. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. t  -.  t  =  u   &    |-  -.  A. t  -.  t  =  v   &    |-  -.  A. u  -.  u  =  x   &    |-  -.  A. u  -.  u  =  w   &    |-  -.  A. v  -.  v  =  x   &    |-  -.  A. v  -.  v  =  w   &    |-  -.  A. x  -.  x  =  t   &    |-  -.  A. x  -.  x  =  v   &    |-  -.  A. w  -.  w  =  t   &    |-  -.  A. w  -.  w  =  v   &    |-  -.  A. w  -.  w  =  y   &    |-  -.  A. v  -.  v  =  z   =>    |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )
 
Theoremax9vax9 28426* Derive ax9 1891 (which has no distinct variable requirement) from a weaker version that requires that its two variables be distinct. The weaker version is axiom scheme B7 of [Tarski] p. 75. The hypotheses are the instances of the weaker version that we need. Neither ax9 1891 nor sp 1717 (which can be derived from ax9 1891) is used by the proof.

Revised on 7-Aug-2015 to remove the dependence on ax-10 2083.

See also the remarks for ax9v 1638 and ax9 1891. This theorem does not actually use ax9v 1638 so that other paths to ax9 1891 can be demonstrated (such as in ax9sep 28428). Theorem ax9 1891 uses this one to make the derivation from ax9v 1638. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  -.  A. t  -.  t  =  u   &    |-  -.  A. t  -.  t  =  z   &    |-  -.  A. u  -.  u  =  x   &    |-  -.  A. u  -.  u  =  w   &    |-  -.  A. z  -.  z  =  x   &    |-  -.  A. z  -.  z  =  w   &    |-  -.  A. x  -.  x  =  t   &    |-  -.  A. x  -.  x  =  z   &    |-  -.  A. w  -.  w  =  t   &    |-  -.  A. w  -.  w  =  z   &    |-  -.  A. w  -.  w  =  y   &    |-  -.  A. x  -.  x  =  v   &    |-  -.  A. v  -.  v  =  y   =>    |- 
 -.  A. x  -.  x  =  y
 
Theoremax9OLD 28427 Theorem showing that ax9 1891 follows from the weaker version ax9v 1638.

See also ax9 1891 for a slightly more direct proof (using lemmas for ax10 1886 derivation).

This theorem normally should not be referenced in any later proof. Instead, the use of ax9 1891 below is preferred, since it is easier to work with (it has no distinct variable conditions) and it is the standard version we have adopted. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  -.  A. x  -.  x  =  y
 
Theoremax9sep 28428 Show that the Separation Axiom ax-sep 4143 and Extensionality ax-ext 2266 implies ax9 1891. Note that ax9 1891 and sp 1717 (which can be derived from ax9 1891) are not used by the proof. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. x  -.  x  =  y
 
18.25.2  Miscellanea
 
Theoremcnaddcom 28429 Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theoremtoycom 28430* Show the commutative law for an operation  O on a toy structure class  C of commuatitive operations on  CC. This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of  C. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
 |-  C  =  { g  e.  Abel  |  ( Base `  g )  =  CC }   &    |-  .+  =  ( +g  `  K )   =>    |-  ( ( K  e.  C  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A 
 .+  B )  =  ( B  .+  A ) )
 
TheoremlubunNEW 28431 The LUB of a union. (Contributed by NM, 5-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  U  =  ( lub `  K )   =>    |-  ( ( K  e.  CLat  /\  S  C_  B  /\  T  C_  B )  ->  ( U `  ( S  u.  T ) )  =  ( ( U `
  S )  .\/  ( U `  T ) ) )
 
18.25.3  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 28432 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 28433 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 28434* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
 |- LSAtoms  =  ( w  e.  _V  |->  ran  (  v  e.  (
 ( Base `  w )  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  { v }
 ) ) )
 
Definitiondf-lshyp 28435* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
 |- LSHyp  =  ( w  e.  _V  |->  { s  e.  ( LSubSp `  w )  |  (
 s  =/=  ( Base `  w )  /\  E. v  e.  ( Base `  w ) ( (
 LSpan `  w ) `  ( s  u.  { v } ) )  =  ( Base `  w )
 ) } )
 
Theoremlshpset 28436* The set of all hyperplanes of a left module or left vector space. The vector  v is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   =>    |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u. 
 { v } )
 )  =  V ) } )
 
Theoremislshp 28437* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   =>    |-  ( W  e.  X  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v }
 ) )  =  V ) ) )
 
Theoremislshpsm 28438* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `
  { v }
 ) )  =  V ) ) )
 
Theoremlshplss 28439 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlshpne 28440 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  U  =/=  V )
 
Theoremlshpnel 28441 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  -.  X  e.  U )
 
Theoremlshpnelb 28442 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N ` 
 { X } )
 )  =  V ) )
 
Theoremlshpnel2N 28443 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
 
Theoremlshpne0 28444 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  X  =/=  .0.  )
 
Theoremlshpdisj 28445 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  ( U  i^i  ( N `
  { X }
 ) )  =  {  .0.  } )
 
Theoremlshpcmp 28446 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
 |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  H )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
TheoremlshpinN 28447 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
 |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  H )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  ( ( T  i^i  U )  e.  H  <->  T  =  U ) )
 
Theoremlsatset 28448* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  X  ->  A  =  ran  (  v  e.  ( V  \  {  .0.  } )  |->  ( N `  { v } ) ) )
 
Theoremislsat 28449* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V 
 \  {  .0.  }
 ) U  =  ( N `  { x } ) ) )
 
Theoremlsatlspsn2 28450 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 28451 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( N `  { X } )  e.  A )
 
Theoremlsatlspsn 28451 The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( N ` 
 { X } )  e.  A )
 
Theoremislsati 28452* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( W  e.  X  /\  U  e.  A ) 
 ->  E. v  e.  V  U  =  ( N ` 
 { v } )
 )
 
Theoremlsateln0 28453* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
 
Theoremlsatlss 28454 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  LMod  ->  A  C_  S )
 
Theoremlsatlssel 28455 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlsatssv 28456 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  Q  C_  V )
 
Theoremlsatn0 28457 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 22918 analog.) (Contributed by NM, 25-Aug-2014.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  U  =/=  {  .0.  }
 )
 
Theoremlsatspn0 28458 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  X  =/=  .0.  ) )
 
Theoremlsator0sp 28459 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  e.  A  \/  ( N `  { X } )  =  {  .0.  } ) )
 
Theoremlsatssn0 28460 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  Q  C_  U )   =>    |-  ( ph  ->  U  =/=  {  .0.  } )
 
Theoremlsatcmp 28461 If two atoms are comparable, they are equal. (atsseq 22920 analog.) TODO: can lspsncmp 15864 shorten this? (Contributed by NM, 25-Aug-2014.)
 |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  A )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
Theoremlsatcmp2 28462 If an atoms is included in at-most an atom, they are equal. More general version of lsatcmp 28461. TODO: can lspsncmp 15864 shorten this? (Contributed by NM, 3-Feb-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  A )   &    |-  ( ph  ->  ( U  e.  A  \/  U  =  {  .0.  } ) )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
Theoremlsatel 28463 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
 |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  A )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  U  =  ( N `  { X } ) )
 
TheoremlsatelbN 28464 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  ( X  e.  U  <->  U  =  ( N `  { X }
 ) ) )
 
Theoremlsat2el 28465 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  X  =/=  .0.  )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  X  e.  Q )   =>    |-  ( ph  ->  P  =  Q )
 
Theoremlsmsat 28466* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 29262 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  T  =/=  {  .0.  }
 )   &    |-  ( ph  ->  Q  C_  ( T  .(+)  U ) )   =>    |-  ( ph  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
 
TheoremlsatfixedN 28467* Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15876. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Q  =/=  ( N `  { X } ) )   &    |-  ( ph  ->  Q  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  { ( X  .+  z ) }
 ) )
 
Theoremlsmsatcv 28468 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22224 analog.) Explicit atom version of lsmcv 15889. (Contributed by NM, 29-Oct-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ( ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) )
 
Theoremlssatomic 28469* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22931 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  U  =/=  {  .0.  } )   =>    |-  ( ph  ->  E. q  e.  A  q  C_  U )
 
Theoremlssats 28470* The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 22934 analog.) (Contributed by NM, 9-Apr-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  U  =  ( N `
  U. { x  e.  A  |  x  C_  U } ) )
 
Theoremlpssat 28471* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 22936 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T  C.  U )   =>    |-  ( ph  ->  E. q  e.  A  ( q  C_  U  /\  -.  q  C_  T ) )
 
Theoremlrelat 28472* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22937 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T  C.  U )   =>    |-  ( ph  ->  E. q  e.  A  ( T  C.  ( T  .(+)  q ) 
 /\  ( T  .(+)  q )  C_  U )
 )
 
Theoremlssatle 28473* The ordering of two subspaces is determined by the atoms under them. (chrelat3 22944 analog.) (Contributed by NM, 29-Oct-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T  C_  U  <->  A. p  e.  A  ( p  C_  T  ->  p 
 C_  U ) ) )
 
Theoremlssat 28474* Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 22936 analog.) (Contributed by NM, 9-Apr-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  U  e.  S  /\  V  e.  S )  /\  U  C.  V ) 
 ->  E. p  e.  A  ( p  C_  V  /\  -.  p  C_  U )
 )
 
Theoremislshpat 28475* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 28438. (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) ) )
 
Syntaxclcv 28476 Extend class notation with the covering relation for a left module or left vector space.
 class  <oLL
 
Definitiondf-lcv 28477* Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation 
A (  <oLL  `  W ) B is read " B covers  A " or " A is covered by  B " , and it means that  B is larger than  A and there is nothing in between. See lcvbr 28479 for binary relation. (df-cv 22852 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  <oLL  =  ( w  e.  _V  |->  { <. t ,  u >.  |  ( ( t  e.  ( LSubSp `
  w )  /\  u  e.  ( LSubSp `  w ) )  /\  ( t  C.  u  /\  -. 
 E. s  e.  ( LSubSp `
  w ) ( t  C.  s  /\  s  C.  u ) ) ) } )
 
Theoremlcvfbr 28478* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  C  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S )  /\  (
 t  C.  u  /\  -. 
 E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) } )
 
Theoremlcvbr 28479* The covers relation for a left vector space (or a left module). (cvbr 22855 analog.) (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) ) )
 
Theoremlcvbr2 28480* The covers relation for a left vector space (or a left module). (cvbr2 22856 analog.) (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  A. s  e.  S  ( ( T 
 C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
 
Theoremlcvbr3 28481* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  A. s  e.  S  ( ( T 
 C_  s  /\  s  C_  U )  ->  (
 s  =  T  \/  s  =  U )
 ) ) ) )
 
Theoremlcvpss 28482 The covers relation implies proper subset. (cvpss 22858 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  T  C.  U )
 
Theoremlcvnbtwn 28483 The covers relation implies no in-betweenness. (cvnbtwn 22859 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   =>    |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T ) )
 
Theoremlcvntr 28484 The covers relation is not transitive. (cvntr 22865 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  -.  R C U )
 
Theoremlcvnbtwn2 28485 The covers relation implies no in-betweenness. (cvnbtwn2 22860 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  R 
 C.  U )   &    |-  ( ph  ->  U  C_  T )   =>    |-  ( ph  ->  U  =  T )
 
Theoremlcvnbtwn3 28486 The covers relation implies no in-betweenness. (cvnbtwn3 22861 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  R 
 C_  U )   &    |-  ( ph  ->  U  C.  T )   =>    |-  ( ph  ->  U  =  R )
 
Theoremlsmcv2 28487 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22866 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  ( N ` 
 { X } )  C_  U )   =>    |-  ( ph  ->  U C ( U  .(+)  ( N `  { X } ) ) )
 
Theoremlcvat 28488* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 22939 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  E. q  e.  A  ( T  .(+)  q )  =  U )
 
Theoremlsatcv0 28489 An atom covers the zero subspace. (atcv0 22915 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  {  .0.  } C Q )
 
Theoremlsatcveq0 28490 A subspace covered by an atom must be the zero subspace. (atcveq0 22921 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U C Q  <->  U  =  {  .0.  } ) )
 
Theoremlsat0cv 28491 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
 
Theoremlcvexchlem1 28492 Lemma for lcvexch 28497. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T  C.  ( T 
 .(+)  U )  <->  ( T  i^i  U )  C.  U ) )
 
Theoremlcvexchlem2 28493 Lemma for lcvexch 28497. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  ( T  i^i  U )  C_  R )   &    |-  ( ph  ->  R 
 C_  U )   =>    |-  ( ph  ->  ( ( R  .(+)  T )  i^i  U )  =  R )
 
Theoremlcvexchlem3 28494 Lemma for lcvexch 28497. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  C_  R )   &    |-  ( ph  ->  R 
 C_  ( T  .(+)  U ) )   =>    |-  ( ph  ->  (
 ( R  i^i  U )  .(+)  T )  =  R )
 
Theoremlcvexchlem4 28495 Lemma for lcvexch 28497. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C ( T  .(+)  U )
 )   =>    |-  ( ph  ->  ( T  i^i  U ) C U )
 
Theoremlcvexchlem5 28496 Lemma for lcvexch 28497. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  ( T  i^i  U ) C U )   =>    |-  ( ph  ->  T C ( T  .(+)  U ) )
 
Theoremlcvexch 28497 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 22942 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( ( T  i^i  U ) C U  <->  T C ( T 
 .(+)  U ) ) )
 
Theoremlcvp 28498 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22948 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (
 ( U  i^i  Q )  =  {  .0.  }  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv1 28499 Covering property of a subspace plus an atom. (chcv1 22928 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( -.  Q  C_  U  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv2 28500 Covering property of a subspace plus an atom. (chcv2 22929 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U  C.  ( U 
 .(+)  Q )  <->  U C ( U 
 .(+)  Q ) ) )
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