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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax10X 28001 Proof of axiom ax-10 1678 from others, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (See remarks for ax12o10lem1 1635 about why we use ax-9v 1632 instead of ax-9 1684.)

Our current practice is to use axiom ax-10 1678 from here on instead of theorem ax10 1677, in order to preferentially use ax-9 1684 instead of ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theorema16gALTX 28002* Alternate proof of a16g 2000 without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
TheoremalequcomX 28003 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremalequcomsX 28004 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
TheoremnalequcomsX 28005 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  -> 
 ph )
 
16.23.4  Derive ax-9 from the weaker version ax-9v
 
Theoremax9X 28006 Theorem showing that ax-9 1684 follows from the weaker version ax-9v 1632.

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, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (New usage is discouraged.)

 |-  -.  A. x  -.  x  =  y
 
16.23.5  Obsolete experiments to study ax-12o
 
Theoremax12-2 28007 Possible alternative to ax-12 1633. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  -.  z  =  y  ->  ( -. 
 A. z  -.  x  =  y  ->  A. z  x  =  y )
 )
 
Theoremax12-3 28008 An equivalent to ax12-2 28007. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. z  -.  x  =  y  ->  ( E. z  x  =  y  ->  E. x  z  =  y ) )
 
Theoremax12OLD 28009 Derive ax-12 1633 from ax-12o 1664. (Contributed by NM, 29-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. z  -.  A. x  -.  z  =  y 
 ->  ( x  =  y 
 ->  A. z  x  =  y ) )
 
Theoremax12-4 28010 Study of candidate for ax-12 1633. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z  -.  x  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
 )
 
Theoremanandii 28011 Elimination of dependent conjuncts. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ch )   &    |-  ( ps  ->  th )   =>    |-  ( ( ( ch 
 /\  ps )  /\  ( ph  /\  th ) )  <-> 
 ( ph  /\  ps )
 )
 
Theoremax12conj2 28012* Conjectured alternative to ax-12 1633. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  z  =  y  ->  ( x  =  y 
 ->  A. z  x  =  y ) )   =>    |-  ( ( -. 
 A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
 )  \/  ( -.  z  =  y  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremhbae-x12 28013* Experiment to study ax-12o 1664. Weak version of hbae 1840. Does not use ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allows ax-9v 1632. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
 
Theoremhbnae-x12 28014* Experiment to study ax-12o 1664. Weak version of hbnae 1844. Does not use ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allows ax-9v 1632. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  A. y  -.  A. x  x  =  y )
 
Theorema12stdy1-x12 28015* Part of a study related to ax-12o 1664. Weak version of a12stdy1 28030. Does not use ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allows ax-9v 1632. The consequent introduces a new variable  z. (Contributed by NM, 7-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  E. x  y  =  z )
 
Theorema12stdy2-x12 28016* Part of a study related to ax-12o 1664. Weak version of a12stdy2 28031. Does not use ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allows ax-9v 1632. The consequent is quantified with a different variable. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  /\  x  =  y )  ->  A. y  y  =  x )
 
Theoremequsexv-x12 28017* Weaker version of equsex 1852 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o but allowing ax-9v 1632. Experiment to study ax-12o 1664. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theoremequvinv 28018* Similar to equvini 1879 without using ax-12o 1664. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequveliv 28019* Similar to equveli 1880 without using ax-12o 1664. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  ->  z  =  y )  <->  x  =  y
 )
 
Theoremequvelv 28020* Similar to equveli 1880 without using ax-12o 1664. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 ) 
 <->  x  =  y )
 
Theorema12study4 28021* Experiment to study ax-12o 1664. The first hypothesis is a conjectured ax-12o 1664 replacement (see ax12 1881 for its derivation from ax-12o 1664). The second hypothesis needs to be proved without using ax-12o 1664, if that is possible. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  z  =  y  ->  ( y  =  x 
 ->  A. z  y  =  x ) )   &    |-  ( -.  A. z  z  =  y  ->  ( (
 z  =  y  /\  y  =  x )  ->  A. z ( -.  z  =  x  ->  y  =  x )
 ) )   =>    |-  ( -.  A. z  z  =  y  ->  ( y  =  x  ->  A. z  y  =  x ) )
 
Theorema12study6 28022* Experiment to study ax-12o 1664 (Contributed by NM, 6-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  z  =  y  ->  ( x  =  y 
 ->  A. z  x  =  y ) )   &    |-  ( -.  z  =  y  ->  ( -.  x  =  y  ->  A. z  -.  x  =  y )
 )   =>    |-  ( -.  z  =  y  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) )
 
Theorema12study8 28023* Experiment to study ax-12o 1664. Closed form of ax12conj2 28012. (Contributed by NM, 6-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
 ) 
 <->  ( ( -.  A. z  z  =  y  ->  ( x  =  y 
 ->  A. z  x  =  y ) )  \/  ( -.  z  =  y  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) ) ) )
 
Theorema12study9 28024* Experiment to study ax-12o 1664. (Contributed by NM, 6-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( -.  A. z  z  =  y  ->  ( x  =  y 
 ->  A. z  x  =  y ) )  <->  A. z ( -.  z  =  y  ->  ( -.  A. z  -.  x  =  y  ->  A. z  x  =  y ) ) )
 
Theorema12peros 28025* Experiment to study ax-12o 1664. (Contributed by NM, 9-Dec-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( -.  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
 ) )   &    |-  ( ps  <->  ( -.  z  =  y  ->  ( -.  x  =  y  ->  A. z  -.  x  =  y ) ) )   &    |-  ( ch  <->  ( -.  z  =  y  ->  ( E. z  x  =  y  ->  A. z  x  =  y ) ) )   &    |-  ( th  <->  ( E. z  -.  z  =  y  ->  ( x  =  y 
 ->  A. z  x  =  y ) ) )   &    |-  ( ta  <->  ( E. z  -.  z  =  y  ->  ( E. z  x  =  y  ->  A. z  x  =  y )
 ) )   =>    |-  ( ph  <->  ( ch  \/  th ) )
 
Theorema12study5rev 28026* Experiment to study ax-12o 1664. The hypothesis is a conjectured ax-12o 1664 replacement. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. y  -.  z  =  x  ->  ( -. 
 A. z  -.  x  =  y  ->  A. z  x  =  y )
 )   =>    |-  ( -.  A. z  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) )
 
Theoremax10lem17ALT 28027* Lemma for ax10 1677. Similar to dveeq2 1928, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. Direct proof of ax10lem25 1674, bypassing ax10lem24 1673 to investigate possible simplifications. Uses ax-12o 1664. (Contributed by NM, 20-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )
 
Theoremax10lem18ALT 28028* Distinctor with bound variable change without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. Uses ax-12o 1664. (Contributed by NM, 22-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. y  y  =  x  ->  ( A. x  x  =  w  ->  A. y  y  =  x ) )
 
TheoremdvelimfALT2 28029* Proof of dvelimh 1974 using dveeq2 1928 (shown as the last hypothesis) instead of ax-12o 1664. As a consequence, theorem a12study2 28038 shows that ax-12o 1664 could be replaced by dveeq2 1928 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theorema12stdy1 28030 Part of a study related to ax-12o 1664. The consequent introduces a new variable  z. There are no distinct variable restrictions. (Contributed by NM, 14-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  E. x  y  =  z )
 
Theorema12stdy2 28031 Part of a study related to ax-12o 1664. The consequent is quantified with a different variable. 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. y  y  =  x )
 
Theorema12stdy3 28032 Part of a study related to ax-12o 1664. 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 28033 Part of a study related to ax-12o 1664. The second antecedent of ax-12o 1664 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 28034 Proof of first hypothesis of a12study 28036. (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 28035 Proof of second hypothesis of a12study 28036. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y )
 
Theorema12study 28036 Rederivation of axiom ax-12o 1664 from two shorter formulas, without using ax-12o 1664. See a12lem1 28034 and a12lem2 28035 for the proofs of the hypotheses (using ax-12o 1664). This is the only known breakdown of ax-12o 1664 into shorter formulas. See a12studyALT 28037 for an alternate proof. Note that the proof depends on ax-11o 1940, whose proof ax11o 1939 depends on ax-12o 1664, meaning that we would have to replace ax-11 1624 with ax-11o 1940 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, 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 28037 Alternate proof of a12study 28036, also without using ax-12o 1664. (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 28038* Reprove ax-12o 1664 using dvelimfALT2 28029, showing that ax-12o 1664 can be replaced by dveeq2 1928 (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 28039 Rederivation of axiom ax-12o 1664 from two other formulas, without using ax-12o 1664. See equvini 1879 and equveli 1880 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 1940, whose proof ax11o 1939 depends on ax-12o 1664, meaning that we would have to replace ax-11 1624 with ax-11o 1940 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 28040* 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 28041* 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 28042* 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 28043* 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 28044* 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 28045* 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 28046* 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 28047* 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 28048* Lemma for ax9 1683. Similar to a4im 1867 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 28049* 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 28050* 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 28051* 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 28052* 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 28053* 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 28054* 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 28055* Lemma for ax9 1683. Similar to a4ime 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   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ps )
 
Theoremax9lem13 28056* Lemma for ax9 1683. Similar to cbv3 1874 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 28057* 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 28058* 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 28059* 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 28049 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 28060* Lemma for ax9 1683. Similar to dvelim 2092 with first hypothesis replaced by distinct variable condition, without using ax-4 1692, ax-9 1684, or ax-10 1678. We used ax9lem6 28049 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 28061* Lemma for ax9 1683. Similar to dveeq2 1928, 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 28062* 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 28064). 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 28063 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 28064 Show that the Separation Axiom ax-sep 4038 and Extensionality ax-ext 2234 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
 
16.23.6  Miscellanea
 
Theoremcnaddcom 28065 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 28066* 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 28067 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 ) ) )
 
16.23.7  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 28068 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 28069 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 28070* 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 28071* 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 28072* 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 28073* 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 28074* 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 28075 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 28076 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 28077 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 28078 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 28079 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 28080 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 28081 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 28082 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 28083 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 28084* 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 28085* 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 28086 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 28087 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 28087 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 28088* 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 28089* 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 28090 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 28091 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 28092 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 28093 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 22755 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 28094 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 28095 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 28096 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 28097 If two atoms are comparable, they are equal. (atsseq 22757 analog.) TODO: can lspsncmp 15704 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 28098 If an atoms is included in at-most an atom, they are equal. More general version of lsatcmp 28097. TODO: can lspsncmp 15704 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 28099 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 28100 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 }
 ) ) )
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