HomeHome Metamath Proof Explorer
Theorem List (p. 298 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfsb4OLD7 29701 A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |-  F/ z ph   =>    |-  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
 
TheoremnfsbOLD7 29702* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/ z ph   =>    |- 
 F/ z [ y  /  x ] ph
 
TheoremhbsbOLD7 29703* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  ->  A. z [
 y  /  x ] ph )
 
TheoremnfsbdOLD7 29704* Deduction version of nfsbOLD7 29702. (Contributed by NM, 15-Feb-2013.)
 |-  F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
TheoremdvelimdfOLD7 29705 Deduction form of dvelimfOLD7 29681. This version may be useful if we want to avoid ax-17 1606 and use ax-16 2096 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ z ch )   &    |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( -.  A. x  x  =  y 
 ->  F/ x ch )
 )
 
Theoremsbco2OLD7 29706 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   =>    |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
 
Theoremsbco2dOLD7 29707 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
 
Theoremsbco3OLD7 29708 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
 
TheoremsbcomOLD7 29709 A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
 |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
 
Theoremsb8OLD7 29710 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
 
Theoremsb8eOLD7 29711 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ y ph   =>    |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
 
Theoremsb9iOLD7 29712 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  ->  A. y [ y  /  x ] ph )
 
Theoremsb9OLD7 29713 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
 
TheoremsbhbOLD7 29714* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by NM, 29-May-2009.)
 |-  (
 ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbcom2OLD7 29715* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
 |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07OLD7 29716* Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorem2sb5rfOLD7 29717* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [
 z  /  x ] [ w  /  y ] ph ) )
 
Theorem2sb6rfOLD7 29718* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )
 )
 
Theoremdfsb7OLD7 29719* An alternate definition of proper substitution df-sb 1639. By introducing a dummy variable  z in the definiens, we are able to eliminate any distinct variable restrictions among the variables  x,  y, and  ph of the definiendum. No distinct variable conflicts arise because  z effectively insulates  x from  y. To achieve this, we use a chain of two substitutions in the form of sb5NEW7 29569, first  z for  x then  y for  z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2283. Theorem sb7hOLD7 29721 provides a version where  ph and  z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
 |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7fOLD7 29720* This version of dfsb7OLD7 29719 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1606 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1639 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ z ph   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7hOLD7 29721* This version of dfsb7OLD7 29719 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1606 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1639 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb10fOLD7 29722* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  F/ x ph   =>    |-  ( [ y  /  z ] ph  <->  E. x ( x  =  y  /\  [ x  /  z ] ph ) )
 
Theorem2exsbOLD7 29723* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x E. y ph  <->  E. z E. w A. x A. y ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsbal2OLD7 29724* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( [
 z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
18.27.2  Obsolete experiments to study ax-12o
 
Theoremax12-2 29725 Possible alternative to ax-12 1878. (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 29726 An equivalent to ax12-2 29725. (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 29727 Derive ax-12 1878 from ax12o 1887. (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 29728 Study of candidate for ax-12 1878. (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 29729 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 29730* Conjectured alternative to ax-12 1878. (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 29731* Experiment to study ax12o 1887. Weak version of hbae 1906. Does not use sp 1728, ax9 1902, ax10 1897, or ax12o 1887 but allows ax9v 1645. (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 29732* Experiment to study ax12o 1887. Weak version of hbnae 1908. Does not use sp 1728, ax9 1902, ax10 1897, or ax12o 1887 but allows ax9v 1645. (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 29733* Part of a study related to ax12o 1887. Weak version of a12stdy1 29748. Does not use sp 1728, ax9 1902, ax10 1897, or ax12o 1887 but allows ax9v 1645. 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 29734* Part of a study related to ax12o 1887. Weak version of a12stdy2 29749. Does not use sp 1728, ax9 1902, ax10 1897, or ax12o 1887 but allows ax9v 1645. 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 29735* Weaker version of equsex 1915 without using sp 1728, ax9 1902, ax10 1897, or ax12o but allowing ax9v 1645. Experiment to study ax12o 1887. (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 29736* Similar to equvini 1940 without using ax12o 1887. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequveliv 29737* Similar to equveli 1941 without using ax12o 1887. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  ->  z  =  y )  <->  x  =  y
 )
 
Theoremequvelv 29738* Similar to equveli 1941 without using ax12o 1887. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 ) 
 <->  x  =  y )
 
Theorema12study4 29739* Experiment to study ax12o 1887. The first hypothesis is a conjectured ax12o 1887 replacement (see ax12 1888 for its derivation from ax12o 1887). The second hypothesis needs to be proved without using ax12o 1887, 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 29740* Experiment to study ax12o 1887 (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 29741* Experiment to study ax12o 1887. Closed form of ax12conj2 29730. (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 29742* Experiment to study ax12o 1887. (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 29743* Experiment to study ax12o 1887. (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 29744* Experiment to study ax12o 1887. The hypothesis is a conjectured ax12o 1887 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 29745* Lemma for ax10 1897. Similar to dveeq2 1893, without using sp 1728, ax9 1902, or ax10 1897 but allowing ax9v 1645. Direct proof of dveeq2 1893, bypassing dvelimnf 1970 to investigate possible simplifications. Uses ax12o 1887. (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 29746* Distinctor with bound variable change without using sp 1728, ax9 1902, or ax10 1897 but allowing ax9v 1645. Uses ax12o 1887. (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 ) )
 
TheoremdvelimfALT2OLD 29747* Proof of dvelimh 1917 using dveeq2 1893 (shown as the last hypothesis) instead of ax12o 1887. As a consequence, theorem a12study2 29756 shows that ax12o 1887 could be replaced by dveeq2 1893 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) Obsolete as of 1-Aug-2017 - NM.
 |-  ( 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 29748 Part of a study related to ax12o 1887. 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 29749 Part of a study related to ax12o 1887. 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 29750 Part of a study related to ax12o 1887. 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 29751 Part of a study related to ax12o 1887. The second antecedent of ax12o 1887 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 29752 Proof of first hypothesis of a12study 29754. (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 29753 Proof of second hypothesis of a12study 29754. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y )
 
Theorema12study 29754 Rederivation of axiom ax12o 1887 from two shorter formulas, without using ax12o 1887. See a12lem1 29752 and a12lem2 29753 for the proofs of the hypotheses (using ax12o 1887). This is the only known breakdown of ax12o 1887 into shorter formulas. See a12studyALT 29755 for an alternate proof. Note that the proof depends on ax11o 1947, whose proof ax11o 1947 depends on ax12o 1887, meaning that we would have to replace ax-11 1727 with ax11o 1947 in an axiomatization that uses the hypotheses in place of ax12o 1887. 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 29755 Alternate proof of a12study 29754, also without using ax12o 1887. (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 29756* Reprove ax12o 1887 using dvelimhw 1747, showing that ax12o 1887 can be replaced by dveeq2 1893 (whose needed instances are the hypotheses here) if we allow distinct variables in axioms other than ax-17 1606. (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 29757 Rederivation of axiom ax12o 1887 from two other formulas, without using ax12o 1887. See equvini 1940 and equveli 1941 for the proofs of the hypotheses (using ax12o 1887). Although the second hypothesis (when expanded to primitives) is longer than ax12o 1887, an open problem is whether it can be derived without ax12o 1887 or from a simpler axiom.

Note also that the proof depends on ax11o 1947, whose proof ax11o 1947 depends on ax12o 1887, meaning that we would have to replace ax-11 1727 with ax11o 1947 in an axiomatization that uses the hypotheses in place of ax12o 1887. 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 29758* Experiment to study ax12o 1887. (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 29759* Experiment to study ax12o 1887. (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 29760* Experiment to study ax12o 1887. (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 29761* Experiment to study ax12o 1887. (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 29762* Lemma for ax9 1902. Similar to equcomi 1664, without using sp 1728, ax9 1902, or ax10 1897. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. w  -.  w  =  x   =>    |-  ( x  =  y 
 ->  y  =  x )
 
Theoremax9lem2 29763* Lemma for ax9 1902. Similar to equequ2 1669, without using sp 1728, ax9 1902, or ax10 1897. (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 29764* Lemma for ax9 1902. Similar to sp 1728, without using sp 1728, ax9 1902, or ax10 1897. (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 29765* Lemma for ax9 1902. Similar to ax9o 1903, without using sp 1728, ax9 1902, or ax10 1897. (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 29766* Lemma for ax9 1902. Similar to spim 1928 with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. (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 29767* Lemma for ax9 1902. 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 29768* Lemma for ax9 1902. Similar to hba1 1731, without using sp 1728, ax9 1902, or ax10 1897. (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 29769* Lemma for ax9 1902. Similar to hbn 1732, without using sp 1728, ax9 1902, or ax10 1897. (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 29770* Lemma for ax9 1902. Similar to hbimd 1733, without using sp 1728, ax9 1902, or ax10 1897. (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 29771* Lemma for ax9 1902. Similar to hban 1748, without using sp 1728, ax9 1902, or ax10 1897. (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 29772* Lemma for ax9 1902. Similar to exlimih 1741, without using sp 1728, ax9 1902, or ax10 1897. (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 29773* Lemma for ax9 1902. Similar to spime 1929 with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. (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 29774* Lemma for ax9 1902. Similar to cbv3 1935 with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. (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 29775* Change bound variable without using sp 1728, ax9 1902, or ax10 1897. (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 29776* Change free variable without using sp 1728, ax9 1902, or ax10 1897. (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 29777* Lemma for ax9 1902. Similar to ax10 1897 but with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. We used ax9lem6 29767 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 29778* Lemma for ax9 1902. Similar to dvelim 1969 with first hypothesis replaced by distinct variable condition, without using sp 1728, ax9 1902, or ax10 1897. We used ax9lem6 29767 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 29779* Lemma for ax9 1902. Similar to dveeq2 1893, without using sp 1728, ax9 1902, or ax10 1897. (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 29780* Derive ax9 1902 (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 1902 nor sp 1728 (which can be derived from ax9 1902) is used by the proof.

Revised on 7-Aug-2015 to remove the dependence on ax10 1897.

See also the remarks for ax9v 1645 and ax9 1902. This theorem does not actually use ax9v 1645 so that other paths to ax9 1902 can be demonstrated (such as in ax9sep 29782). Theorem ax9 1902 uses this one to make the derivation from ax9v 1645. (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 29781 Theorem showing that ax9 1902 follows from the weaker version ax9v 1645.

See also ax9 1902 for a slightly more direct proof (using lemmas for ax10 1897 derivation).

This theorem normally should not be referenced in any later proof. Instead, the use of ax9 1902 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 29782 Show that the Separation Axiom ax-sep 4157 and Extensionality ax-ext 2277 implies ax9 1902. Note that ax9 1902 and sp 1728 (which can be derived from ax9 1902) are not used by the proof. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  A. x  -.  x  =  y
 
18.27.3  Miscellanea
 
Theoremcnaddcom 29783 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 29784* 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 29785 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 ) ) )
 
18.27.4  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 29786 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 29787 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 29788* 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 29789* 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 29790* 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 29791* 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 29792* 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 29793 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 29794 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 29795 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 29796 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 29797 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 29798 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 29799 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 29800 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 ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >