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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-inftyexpiinj 33901 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33900 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
 |-  (
 ( A  e.  ( -u pi (,] pi ) 
 /\  B  e.  ( -u pi (,] pi ) )  ->  ( A  =  B  <->  (inftyexpi  `  A )  =  (inftyexpi  `  B ) ) )
 
Theorembj-inftyexpidisj 33902 An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
 |-  -.  (inftyexpi  `
  A )  e. 
 CC
 
Syntaxcccinfty 33903 Syntax for the circle at infinity CCinfty.
 class CCinfty
 
Definitiondf-bj-ccinfty 33904 Definition of the circle at infinity CCinfty. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
 |- CCinfty  =  ran inftyexpi
 
Theorembj-ccinftydisj 33905 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
 |-  ( CC  i^i CCinfty )  =  (/)
 
Theorembj-elccinfty 33906 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  e. CCinfty )
 
Syntaxcccbar 33907 Syntax for the set of extended complex numbers CCbar.
 class CCbar
 
Definitiondf-bj-ccbar 33908 Definition of the set of extended complex numbers CCbar. (Contributed by BJ, 22-Jun-2019.)
 |- CCbar  =  ( CC  u. CCinfty )
 
Theorembj-ccssccbar 33909 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |-  CC  C_ CCbar
 
Theorembj-ccinftyssccbar 33910 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |- CCinfty  C_ CCbar
 
Syntaxcpinfty 33911 Syntax for pinfty.
 class pinfty
 
Definitiondf-bj-pinfty 33912 Definition of pinfty. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  =  (inftyexpi  `  0 )
 
Theorembj-pinftyccb 33913 The class pinfty is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  e. CCbar
 
Theorembj-pinftynrr 33914 The extended complex number pinfty is not a complex number. (Contributed by BJ, 27-Jun-2019.)
 |-  -. pinfty  e. 
 CC
 
Syntaxcminfty 33915 Syntax for minfty.
 class minfty
 
Definitiondf-bj-minfty 33916 Definition of minfty. (Contributed by BJ, 27-Jun-2019.)
 |- minfty  =  (inftyexpi  `  pi )
 
Theorembj-minftyccb 33917 The class minfty is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
 |- minfty  e. CCbar
 
Theorembj-minftynrr 33918 The extended complex number minfty is not a complex number. (Contributed by BJ, 27-Jun-2019.)
 |-  -. minfty  e. 
 CC
 
Theorembj-pinftynminfty 33919 The extended complex numbers pinfty and minfty are different. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  =/= minfty
 
Syntaxcrrbar 33920 Syntax for the set of extended real numbers RRbar.
 class RRbar
 
Definitiondf-bj-rrbar 33921 Definition of the set of extended real numbers RRbar. See df-xr 9633. (Contributed by BJ, 29-Jun-2019.)
 |- RRbar  =  ( RR  u.  {minfty , pinfty } )
 
Syntaxcinfty 33922 Syntax for infty.
 class infty
 
Definitiondf-bj-infty 33923 Definition of infty, the point at infinity of the real or complex projective line. (Contributed by BJ, 27-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
 |- infty  =  ~P U. CC
 
Syntaxccchat 33924 Syntax for CChat.
 class CChat
 
Definitiondf-bj-cchat 33925 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
 |- CChat  =  ( CC  u.  {infty } )
 
Syntaxcrrhat 33926 Syntax for RRhat.
 class RRhat
 
Definitiondf-bj-rrhat 33927 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
 |- RRhat  =  ( RR  u.  {infty } )
 
Theorembj-rrhatsscchat 33928 The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
 |- RRhat  C_ CChat
 
21.29.6.3  Addition and opposite

We define the operations on the extended real and complex numbers and on the real and complex projective lines simultaneously, thus "overloading" the operations.

 
Syntaxcaddcc 33929 Syntax for the addition of extended complex numbers.
 class +cc
 
Definitiondf-bj-addc 33930 Define the additions on the extended complex numbers (on the subset of  (CCbar  X. CCbar ) where it makes sense) and on the complex projective line (Riemann sphere). (Contributed by BJ, 22-Jun-2019.)
 |- +cc  =  ( x  e.  ( ( ( CC  X. CCbar )  u.  (CCbar  X.  CC )
 )  u.  ( (CChat  X. CChat )  u.  (Diag ` CCinfty ) ) )  |->  if (
 ( ( 1st `  x )  = infty  \/  ( 2nd `  x )  = infty ) , infty ,  if ( ( 1st `  x )  e.  CC ,  if (
 ( 2nd `  x )  e.  CC ,  ( ( 1st `  x )  +  ( 2nd `  x ) ) ,  ( 2nd `  x ) ) ,  ( 1st `  x ) ) ) )
 
Syntaxcoppcc 33931 Syntax for the opposite of extended complex numbers.
 class -cc
 
Definitiondf-bj-oppc 33932 Define the negation (operation givin the opposite) the set of extended complex numbers and the complex projective line (Riemann sphere). One could use the prcpal function in the infinite case, but we want to use only basic functions at this point. (Contributed by BJ, 22-Jun-2019.)
 |- -cc  =  ( x  e.  (CCbar  u. CChat ) 
 |->  if ( x  = infty , infty ,  if ( x  e.  CC ,  -u x ,  (inftyexpi  `  if ( 0  <  ( 1st `  x ) ,  ( ( 1st `  x )  -  pi ) ,  ( ( 1st `  x )  +  pi ) ) ) ) ) )
 
21.29.6.4  Argument, multiplication and inverse

Since one needs arguments in order to define multiplication in CCbar, it seems harder to put this at the very beginning of the introduction of complex numbers.

 
Syntaxcprcpal 33933 Syntax for the function prcpal.
 class prcpal
 
Definitiondf-bj-prcpal 33934 Define the function prcpal. (Contributed by BJ, 22-Jun-2019.)
 |- prcpal  =  ( x  e.  RR  |->  ( ( x  mod  (
 2  x.  pi ) )  -  if (
 ( x  mod  (
 2  x.  pi ) )  <_  pi , 
 0 ,  ( 2  x.  pi ) ) ) )
 
Syntaxcarg 33935 Syntax for the argument of a nonzero extended complex number.
 class Arg
 
Definitiondf-bj-arg 33936 Define the argument of a nonzero extended complex number. By convention, it has values in  ( -u pi ,  pi ]. Another convention chooses  [ 0 ,  2 pi ) but the present one simplifies formulas giving the argument as an arc-tangent. (Contributed by BJ, 22-Jun-2019.)
 |- Arg  =  ( x  e.  (CCbar  \  {
 0 } )  |->  if ( x  e.  CC ,  ( Im `  ( log `  x ) ) ,  ( 1st `  x ) ) )
 
Syntaxcmulc 33937 Syntax for the multiplication of extended complex numbers.
 class .cc
 
Definitiondf-bj-mulc 33938 Define the multiplication of extended complex numbers and of the complex projective line (Riemann sphere). In our convention, a product with 0 is 0, even when the other factor is infinite. An alternate convention leaves products of 0 with an infinite number undefined since the multiplication is not continuous at these points. Note that our convention entails  ( 0  /  0 )  =  0. (Contributed by BJ, 22-Jun-2019.)
 |- .cc  =  ( x  e.  ( (CCbar  X. CCbar )  u.  (CChat  X. CChat ) )  |->  if ( ( ( 1st `  x )  =  0  \/  ( 2nd `  x )  =  0 ) ,  0 ,  if ( ( ( 1st `  x )  = infty  \/  ( 2nd `  x )  = infty ) , infty ,  if ( x  e.  ( CC  X.  CC ) ,  ( ( 1st `  x )  x.  ( 2nd `  x ) ) ,  (inftyexpi  `  (prcpal `  ( (Arg `  ( 1st `  x ) )  +  (Arg `  ( 2nd `  x ) ) ) ) ) ) ) ) )
 
Syntaxcinvc 33939 Syntax for the inverse of nonzero extended complex numbers.
 class invc
 
Definitiondf-bj-invc 33940 Define inversion, which maps a nonzero extended complex number or element of the complex projective line (Riemann sphere) to its inverse. Beware of the overloading: the equality  (invc `  0 )  = infty is to be understood in the complex projective line, but 0 as an extended complex number does not have an inverse, which we can state as  (invc `  0
)  e/ CCbar. (Contributed by BJ, 22-Jun-2019.)
 |- invc  =  ( x  e.  (CCbar  u. CChat ) 
 |->  if ( x  =  0 , infty ,  if ( x  e.  CC ,  ( 1  /  x ) ,  0 )
 ) )
 
21.29.7  Monoids

See ccmn 16613 and subsequents. The first few statements of this subsection can be put very early after ccmn 16613. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabl 16614 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.

 
Theorembj-cmnssmnd 33941 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |- CMnd  C_  Mnd
 
Theorembj-cmnssmndel 33942 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 16628, which relies on iscmn 16620. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e. CMnd  ->  A  e.  Mnd )
 
Theorembj-ablssgrp 33943 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  Abel  C_ 
 Grp
 
Theorembj-ablssgrpel 33944 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 16618. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  Abel  ->  A  e.  Grp )
 
Theorembj-ablsscmn 33945 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  Abel  C_ CMnd
 
Theorembj-ablsscmnel 33946 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 16619. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  Abel  ->  A  e. CMnd )
 
Theorembj-modssabl 33947 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 17369; see also lmodgrp 17331 and lmodcmn 17370.) (Contributed by BJ, 9-Jun-2019.)
 |-  LMod  C_ 
 Abel
 
Theorembj-vecssmod 33948 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  LVec  C_ 
 LMod
 
Theorembj-vecssmodel 33949 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 17564. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  LVec  ->  A  e.  LMod )
 
21.29.7.1  Finite sums in monoids

UPDATE: a similar summation is already defined as df-gsum 14701 (although it mixes finite and infinite sums, which makes it harder to understand).

 
Syntaxcfinsum 33950 Syntax for the class "finite summation in monoids".
 class FinSum
 
Definitiondf-bj-finsum 33951* Finite summation in commutative monoids. This finite summation function can be extended to pairs 
<. y ,  z >. where  y is a left-unital magma and  z is defined on a totally ordered set (choosing left-associative composition), or dropping unitality and requiring nonempty families, or on any monoids for families of permutable elements, etc. We use the term "summation", even though the definition stands for any unital, commutative and associative composition law. (Contributed by BJ, 9-Jun-2019.)
 |- FinSum  =  ( x  e.  { <. y ,  z >.  |  ( y  e. CMnd  /\  E. t  e.  Fin  z : t --> ( Base `  y )
 ) }  |->  ( iota
 s E. m  e. 
 NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  ( 
 seq 1 ( (
 +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x ) `  ( f `  n ) ) ) ) `  m ) ) ) )
 
Theorembj-finsumval0 33952* Value of a finite sum. (Contributed by BJ, 9-Jun-2019.)
 |-  ( ph  ->  A  e. CMnd )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  B : I --> ( Base `  A ) )   =>    |-  ( ph  ->  ( A FinSum  B )  =  (
 iota s E. m  e.  NN0  E. f ( f : ( 1
 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `
  ( f `  n ) ) ) ) `  ( # `  I ) ) ) ) )
 
21.29.8  Affine, Euclidean, and Cartesian geometry

A few basic theorems to start affine, Euclidean, and Cartesian geometry.

 
21.29.8.1  Convex hull in real vector spaces

A few basic definitions and theorems about convex hulls in real vector spaces. TODO: prove inclusion in the class of subcomplex vector spaces.

 
Syntaxcrrvec 33953 Syntax for the class of real vector spaces.
 class RR-Vec
 
Definitiondf-bj-rrvec 33954 Definition of the class of real vector spaces. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  =  { x  e.  LVec  |  (Scalar `  x )  = RRfld }
 
Theorembj-rrvecssvec 33955 Real vector spaces are vector spaces. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  C_  LVec
 
Theorembj-rrvecssvecel 33956 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 9-Jun-2019.)
 |-  ( A  e. RR-Vec  ->  A  e.  LVec
 )
 
Theorembj-rrvecsscmn 33957 (The additive groups of) real vector spaces are commutative monoids. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  C_ CMnd
 
Theorembj-rrvecsscmnel 33958 (The additive groups of) real vector spaces are commutative monoids (elemental version). (Contributed by BJ, 9-Jun-2019.)
 |-  ( A  e. RR-Vec  ->  A  e. CMnd )
 
21.29.8.2  Complex numbers (supplements)

Some lemmas to ease algebraic manipulations.

 
Theorembj-subcom 33959 A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  -  ( B  x.  A ) )  =  0 )
 
Theorembj-lsub 33960 Left-subtraction. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  =  C  <->  A  =  ( C  -  B ) ) )
 
Theorembj-rsub 33961 Right-subtraction. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  =  C  <->  B  =  ( C  -  A ) ) )
 
Theorembj-msub 33962 A subtraction law. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  =  ( C  -  B )  <->  B  =  ( C  -  A ) ) )
 
Theorembj-ldiv 33963 Left-division. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  =  C  <->  A  =  ( C  /  B ) ) )
 
Theorembj-rdiv 33964 Right-division. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  =  C  <->  B  =  ( C  /  A ) ) )
 
Theorembj-mdiv 33965 A division law. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A  =  ( C  /  B )  <->  B  =  ( C  /  A ) ) )
 
Theorembj-lineq 33966 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  (
 ( ( A  x.  X )  +  B )  =  Y  <->  X  =  (
 ( Y  -  B )  /  A ) ) )
 
Theorembj-lineqi 33967 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  ( ( A  x.  X )  +  B )  =  Y )   =>    |-  ( ph  ->  X  =  ( ( Y  -  B )  /  A ) )
 
21.29.8.3  Barycentric coordinates

Lemmas about barycentric coordinates. For the moment, this is limited to the one-dimensional case (complex line), where existence and uniqueness of barycentric coordinates is proved by bj-bary1 33970 (which computes them).

 
Theorembj-bary1lem 33968 A lemma for barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  X  =  ( ( ( ( B  -  X ) 
 /  ( B  -  A ) )  x.  A )  +  (
 ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )
 
Theorembj-bary1lem1 33969 Existence and uniqueness (and actual computation) of barycentric coordinates in dimension 1 (complex line). (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  T  e.  CC )   =>    |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  T  =  ( ( X  -  A ) 
 /  ( B  -  A ) ) ) )
 
Theorembj-bary1 33970 Barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  T  e.  CC )   =>    |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  <->  ( S  =  ( ( B  -  X ) 
 /  ( B  -  A ) )  /\  T  =  ( ( X  -  A )  /  ( B  -  A ) ) ) ) )
 
21.30  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 33999, means that the definition or theorem is not used for the derivation of hlathil 36978. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 36978.

 
Theoremfsumshftd 33971* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 13561. The proof demonstrates how this can be derived starting from from fsumshft 13561. (Contributed by NM, 1-Nov-2019.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  (
 ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( ( ph  /\  j  =  ( k  -  K ) ) 
 ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K ) ) B )
 
TheoremfsumshftdOLD 33972* Obsolete version of fsumshftd 33971 as of 1-Nov-2019. (Contributed by NM, 1-Nov-2019.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  (
 ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( ( ph  /\  j  =  ( k  -  K ) ) 
 ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K ) ) B )
 
Axiomax-riotaBAD 33973 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse  A. See also comments for df-iota 5551. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6246, CONFLICTS WITH (THE NEW) df-riota 6246 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
 |-  ( iota_ x  e.  A  ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph )
 ) ,  ( Undef ` 
 { x  |  x  e.  A } ) )
 
TheoremriotaclbgBAD 33974* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( A  e.  V  ->  ( E! x  e.  A  ph  <->  (
 iota_ x  e.  A  ph )  e.  A ) )
 
TheoremriotaclbBAD 33975* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  A  e.  _V   =>    |-  ( E! x  e.  A  ph  <->  ( iota_ x  e.  A  ph )  e.  A )
 
Theoremriotasvd 33976* Deduction version of riotasv 33979. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  D  =  (
 iota_ x  e.  A  A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ( ph  /\  A  e.  V )  ->  (
 ( y  e.  B  /\  ps )  ->  D  =  C ) )
 
Theoremriotasv2d 33977* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4653). Special case of riota2f 6268. (Contributed by NM, 2-Mar-2013.)
 |-  F/ y ph   &    |-  ( ph  ->  F/_ y F )   &    |-  ( ph  ->  F/ y ch )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A  A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  y  =  E ) 
 ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  y  =  E )  ->  C  =  F )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  D  =  F )
 
Theoremriotasv2s 33978* The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4653) in the form of a substitution instance. Special case of riota2f 6268. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  D  =  ( iota_ x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C ) )   =>    |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  /  y ]_ C )
 
Theoremriotasv 33979* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4653). Special case of riota2f 6268. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  A  e.  _V   &    |-  D  =  (
 iota_ x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C ) )   =>    |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
 
Theoremriotasv3d 33980* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4653) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ y th )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A  A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  C  =  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E. y  e.  B  ps )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  th )
 
21.30.1  Experiments with weak deduction theorem
 
Theoremelimhyps 33981 A version of elimhyp 3998 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
 |-  [. B  /  x ]. ph   =>    |-  [. if ( ph ,  x ,  B )  /  x ]. ph
 
Theoremdedths 33982 A version of weak deduction theorem dedth 3991 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
 |-  [. if ( ph ,  x ,  B )  /  x ].
 ps   =>    |-  ( ph  ->  ps )
 
TheoremrenegclALT 33983 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 9883. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  -u A  e.  RR )
 
Theoremelimhyps2 33984 Generalization of elimhyps 33981 that is not useful unless we can separately prove  |-  A  e.  _V. (Contributed by NM, 13-Jun-2019.)
 |-  [. B  /  x ]. ph   =>    |-  [. if ( [. A  /  x ]. ph ,  A ,  B )  /  x ]. ph
 
Theoremdedths2 33985 Generalization of dedths 33982 that is not useful unless we can separately prove  |-  A  e.  _V. (Contributed by NM, 13-Jun-2019.)
 |-  [. if ( [. A  /  x ].
 ph ,  A ,  B )  /  x ].
 ps   =>    |-  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
 
21.30.2  Miscellanea
 
Theoremcnaddcom 33986 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 33987* 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 ) )
 
21.30.3  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 33988 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 33989 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 33990* 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 33991* 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 33992* 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 33993* 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 33994* 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 33995 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 33996 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 33997 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 33998 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 33999 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 34000 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.  )
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