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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorema12study2 28301* Reprove ax-12o 1664 using dvelimfALT2 28292, showing that ax-12o 1664 can be replaced by dveeq2 1929 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1628. (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 28302 Rederivation of axiom ax-12o 1664 from two other formulas, without using ax-12o 1664. See equvini 1880 and equveli 1881 for the proofs of the hypotheses (using ax-12o 1664). Although the second hypothesis (when expanded to primitives) is longer than ax-12o 1664, an open problem is whether it can be derived without ax-12o 1664 or from a simpler axiom.

Note also that the proof depends on ax-11o 1941, whose proof ax11o 1940 depends on ax-12o 1664, meaning that we would have to replace ax-11 1624 with ax-11o 1941 in an axiomatization that uses the hypotheses in place of ax-12o 1664. 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 28303* Experiment to study ax-12o 1664. (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 28304* Experiment to study ax-12o 1664. (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 28305* Experiment to study ax-12o 1664. (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 28306* Experiment to study ax-12o 1664. (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 28307* Lemma for ax9 1683. Similar to equcomi 1822, without using ax-4 1692, ax-9 1684, or ax-10 1678. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   =>    |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax9lem2 28308* Lemma for ax9 1683. Similar to equequ2 1830, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28309* Lemma for ax9 1683. Similar to ax4 1691, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28310* Lemma for ax9 1683. Similar to ax9o 1814, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28311* Lemma for ax9 1683. Similar to a4im 1868 with distinct variables, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28312* Lemma for ax9 1683. 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 28313* Lemma for ax9 1683. Similar to hba1 1718, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28314* Lemma for ax9 1683. Similar to hbn 1722, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28315* Lemma for ax9 1683. Similar to hbimd 1809, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28316* Lemma for ax9 1683. Similar to hban 1724, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28317* Lemma for ax9 1683. Similar to exlimih 1782, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28318* Lemma for ax9 1683. Similar to a4ime 1869 with distinct variables, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28319* Lemma for ax9 1683. Similar to cbv3 1875 with distinct variables, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28320* Change bound variable without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28321* Change free variable without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28322* Lemma for ax9 1683. Similar to ax-10 1678 but with distinct variables, without using ax-4 1692, ax-9 1684, or ax-10 1678. We used ax9lem6 28312 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 28323* Lemma for ax9 1683. Similar to dvelim 2096 with first hypothesis replaced by distinct variable condition, without using ax-4 1692, ax-9 1684, or ax-10 1678. We used ax9lem6 28312 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 28324* Lemma for ax9 1683. Similar to dveeq2 1929, without using ax-4 1692, ax-9 1684, or ax-10 1678. (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 28325* Derive ax-9 1684 (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 ax-9 1684 nor ax-4 1692 (which can be derived from ax-9 1684) is used by the proof.

Revised on 7-Aug-2015 to remove the dependence on ax-10 1678.

See also the remarks for ax-9v 1632 and ax9 1683. This theorem does not actually use ax-9v 1632 so that other paths to ax-9 1684 can be demonstrated (such as in ax9sep 28327). Theorem ax9 1683 uses this one to make the derivation from ax-9v 1632. (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 28326 Theorem showing that ax-9 1684 follows from the weaker version ax-9v 1632.

See also ax9 1683 for a slightly more direct proof (using lemmas for ax10 1677 derivation).

This theorem normally should not be referenced in any later proof. Instead, the use of ax-9 1684 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 28327 Show that the Separation Axiom ax-sep 4115 and Extensionality ax-ext 2239 implies ax-9 1684. Note that ax-9 1684 and ax-4 1692 (which can be derived from ax-9 1684) are not used by the proof. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. x  -.  x  =  y
 
18.25.6  Miscellanea
 
Theoremcnaddcom 28328 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 28329* 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 28330 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 ) ) )
 
18.25.7  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 28331 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 28332 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 28333* 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 28334* 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 28335* 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 28336* 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 28337* 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 28338 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 28339 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 28340 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 28341 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 28342 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 28343 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 28344 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 28345 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 28346 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 28347* 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 28348* 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 28349 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 28350 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 28350 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 28351* 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 28352* 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 28353 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 28354 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 28355 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 28356 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 22885 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 28357 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 28358 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 28359 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 28360 If two atoms are comparable, they are equal. (atsseq 22887 analog.) TODO: can lspsncmp 15831 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 28361 If an atoms is included in at-most an atom, they are equal. More general version of lsatcmp 28360. TODO: can lspsncmp 15831 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 28362 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 28363 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 28364 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 28365* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 29161 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 28366* Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15843. (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 28367 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22191 analog.) Explicit atom version of lsmcv 15856. (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 28368* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22898 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 28369* 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 22901 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 28370* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 22903 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 28371* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22904 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 28372* The ordering of two subspaces is determined by the atoms under them. (chrelat3 22911 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 28373* 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 22903 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 28374* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 28337. (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 28375 Extend class notation with the covering relation for a left module or left vector space.
 class  <oLL
 
Definitiondf-lcv 28376* 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 28378 for binary relation. (df-cv 22819 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 28377* 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 28378* The covers relation for a left vector space (or a left module). (cvbr 22822 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 28379* The covers relation for a left vector space (or a left module). (cvbr2 22823 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 28380* 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 28381 The covers relation implies proper subset. (cvpss 22825 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 28382 The covers relation implies no in-betweenness. (cvnbtwn 22826 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 28383 The covers relation is not transitive. (cvntr 22832 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 28384 The covers relation implies no in-betweenness. (cvnbtwn2 22827 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 28385 The covers relation implies no in-betweenness. (cvnbtwn3 22828 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 28386 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22833 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 28387* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 22906 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 28388 An atom covers the zero subspace. (atcv0 22882 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 28389 A subspace covered by an atom must be the zero subspace. (atcveq0 22888 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 28390 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 28391 Lemma for lcvexch 28396. (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 28392 Lemma for lcvexch 28396. (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 28393 Lemma for lcvexch 28396. (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 28394 Lemma for lcvexch 28396. (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 28395 Lemma for lcvexch 28396. (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 28396 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 22909 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 28397 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22915 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 28398 Covering property of a subspace plus an atom. (chcv1 22895 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 28399 Covering property of a subspace plus an atom. (chcv2 22896 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 ) ) )
 
Theoremlsatexch 28400 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 22921 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q 
 C_  ( U  .(+)  R ) )   &    |-  ( ph  ->  ( U  i^i  Q )  =  {  .0.  }
 )   =>    |-  ( ph  ->  R  C_  ( U  .(+)  Q ) )
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