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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorema12stdy3 28501 Part of a study related to ax12o 1875. The consequent introduces two new variables. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  /\  x  =  y )  ->  A. v E. y  x  =  w )
 
Theorema12stdy4 28502 Part of a study related to ax12o 1875. The second antecedent of ax12o 1875 is replaced. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. z  z  =  x  ->  ( A. y  z  =  x  ->  ( x  =  y 
 ->  A. z  x  =  y ) ) )
 
Theorema12lem1 28503 Proof of first hypothesis of a12study 28505. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. z  z  =  y  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
 
Theorema12lem2 28504 Proof of second hypothesis of a12study 28505. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y )
 
Theorema12study 28505 Rederivation of axiom ax12o 1875 from two shorter formulas, without using ax12o 1875. See a12lem1 28503 and a12lem2 28504 for the proofs of the hypotheses (using ax12o 1875). This is the only known breakdown of ax12o 1875 into shorter formulas. See a12studyALT 28506 for an alternate proof. Note that the proof depends on ax11o 1934, whose proof ax11o 1934 depends on ax12o 1875, meaning that we would have to replace ax-11 1715 with ax11o 1934 in an axiomatization that uses the hypotheses in place of ax12o 1875. Whether this can be avoided is an open problem. (Contributed by NM, 15-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 ) ) )
 
Theorema12studyALT 28506 Alternate proof of a12study 28505, also without using ax12o 1875. (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 28507* Reprove ax12o 1875 using dvelimhw 1735, showing that ax12o 1875 can be replaced by dveeq2 1880 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1603. (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 28508 Rederivation of axiom ax12o 1875 from two other formulas, without using ax12o 1875. See equvini 1927 and equveli 1928 for the proofs of the hypotheses (using ax12o 1875). Although the second hypothesis (when expanded to primitives) is longer than ax12o 1875, an open problem is whether it can be derived without ax12o 1875 or from a simpler axiom.

Note also that the proof depends on ax11o 1934, whose proof ax11o 1934 depends on ax12o 1875, meaning that we would have to replace ax-11 1715 with ax11o 1934 in an axiomatization that uses the hypotheses in place of ax12o 1875. 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 28509* Experiment to study ax12o 1875. (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 28510* Experiment to study ax12o 1875. (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 28511* Experiment to study ax12o 1875. (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 28512* Experiment to study ax12o 1875. (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 28513* Lemma for ax9 1889. Similar to equcomi 1646, without using sp 1716, ax9 1889, or ax10 1884. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   =>    |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax9lem2 28514* Lemma for ax9 1889. Similar to equequ2 1649, without using sp 1716, ax9 1889, or ax10 1884. (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 28515* Lemma for ax9 1889. Similar to sp 1716, without using sp 1716, ax9 1889, or ax10 1884. (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 28516* Lemma for ax9 1889. Similar to ax9o 1890, without using sp 1716, ax9 1889, or ax10 1884. (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 28517* Lemma for ax9 1889. Similar to spim 1915 with distinct variables, without using sp 1716, ax9 1889, or ax10 1884. (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 28518* Lemma for ax9 1889. 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 28519* Lemma for ax9 1889. Similar to hba1 1719, without using sp 1716, ax9 1889, or ax10 1884. (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 28520* Lemma for ax9 1889. Similar to hbn 1720, without using sp 1716, ax9 1889, or ax10 1884. (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 28521* Lemma for ax9 1889. Similar to hbimd 1721, without using sp 1716, ax9 1889, or ax10 1884. (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 28522* Lemma for ax9 1889. Similar to hban 1736, without using sp 1716, ax9 1889, or ax10 1884. (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 28523* Lemma for ax9 1889. Similar to exlimih 1729, without using sp 1716, ax9 1889, or ax10 1884. (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 28524* Lemma for ax9 1889. Similar to spime 1916 with distinct variables, without using sp 1716, ax9 1889, or ax10 1884. (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 28525* Lemma for ax9 1889. Similar to cbv3 1922 with distinct variables, without using sp 1716, ax9 1889, or ax10 1884. (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 28526* Change bound variable without using sp 1716, ax9 1889, or ax10 1884. (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 28527* Change free variable without using sp 1716, ax9 1889, or ax10 1884. (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 28528* Lemma for ax9 1889. Similar to ax10 1884 but with distinct variables, without using sp 1716, ax9 1889, or ax10 1884. We used ax9lem6 28518 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 28529* Lemma for ax9 1889. Similar to dvelim 1956 with first hypothesis replaced by distinct variable condition, without using sp 1716, ax9 1889, or ax10 1884. We used ax9lem6 28518 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 28530* Lemma for ax9 1889. Similar to dveeq2 1880, without using sp 1716, ax9 1889, or ax10 1884. (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 28531* Derive ax9 1889 (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 1889 nor sp 1716 (which can be derived from ax9 1889) is used by the proof.

Revised on 7-Aug-2015 to remove the dependence on ax10 1884.

See also the remarks for ax9v 1636 and ax9 1889. This theorem does not actually use ax9v 1636 so that other paths to ax9 1889 can be demonstrated (such as in ax9sep 28533). Theorem ax9 1889 uses this one to make the derivation from ax9v 1636. (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 28532 Theorem showing that ax9 1889 follows from the weaker version ax9v 1636.

See also ax9 1889 for a slightly more direct proof (using lemmas for ax10 1884 derivation).

This theorem normally should not be referenced in any later proof. Instead, the use of ax9 1889 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 28533 Show that the Separation Axiom ax-sep 4141 and Extensionality ax-ext 2264 implies ax9 1889. Note that ax9 1889 and sp 1716 (which can be derived from ax9 1889) are not used by the proof. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. x  -.  x  =  y
 
18.27.2  Miscellanea
 
Theoremcnaddcom 28534 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 28535* 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 28536 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 ) ) )
 
18.27.3  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 28537 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 28538 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 28539* 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 28540* 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 28541* 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 28542* 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 28543* 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 28544 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 28545 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 28546 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 28547 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 28548 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 28549 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 28550 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 28551 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 28552 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 28553* 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 28554* 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 28555 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 28556 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 28556 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 28557* 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 28558* 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 28559 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 28560 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 28561 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 28562 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 22925 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 28563 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 28564 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 28565 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 28566 If two atoms are comparable, they are equal. (atsseq 22927 analog.) TODO: can lspsncmp 15869 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 28567 If an atoms is included in at-most an atom, they are equal. More general version of lsatcmp 28566. TODO: can lspsncmp 15869 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 28568 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 28569 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 28570 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 28571* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 29367 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 28572* Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15881. (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 28573 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22231 analog.) Explicit atom version of lsmcv 15894. (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 28574* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22938 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 28575* 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 22941 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 28576* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 22943 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 28577* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22944 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 28578* The ordering of two subspaces is determined by the atoms under them. (chrelat3 22951 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 28579* 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 22943 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 28580* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 28543. (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 28581 Extend class notation with the covering relation for a left module or left vector space.
 class  <oLL
 
Definitiondf-lcv 28582* 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 28584 for binary relation. (df-cv 22859 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 28583* 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 28584* The covers relation for a left vector space (or a left module). (cvbr 22862 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 28585* The covers relation for a left vector space (or a left module). (cvbr2 22863 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 28586* 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 28587 The covers relation implies proper subset. (cvpss 22865 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 28588 The covers relation implies no in-betweenness. (cvnbtwn 22866 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 28589 The covers relation is not transitive. (cvntr 22872 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 28590 The covers relation implies no in-betweenness. (cvnbtwn2 22867 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 28591 The covers relation implies no in-betweenness. (cvnbtwn3 22868 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 28592 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22873 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 28593* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 22946 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 28594 An atom covers the zero subspace. (atcv0 22922 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 28595 A subspace covered by an atom must be the zero subspace. (atcveq0 22928 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 28596 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 28597 Lemma for lcvexch 28602. (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 28598 Lemma for lcvexch 28602. (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 28599 Lemma for lcvexch 28602. (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 28600 Lemma for lcvexch 28602. (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 )
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