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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-1uplex 32501 A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
 |-  ((| A|)  e.  _V  <->  A  e.  _V )
 
Theorembj-1upln0 32502 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
 |- (| A|)  =/= 
 (/)
 
Syntaxbj-c2uple 32503 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
 class (| A,  B|)
 
Definitiondf-bj-2upl 32504 Definition of the Morse couple. See df-bj-1upl 32491. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 32505, bj-2uplth 32514, bj-2uplex 32515, and the properties of the projections (see df-bj-pr1 32494 and df-bj-pr2 32508). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |- (| A,  B|)  =  ((| A|)  u.  ( { 1o }  X. tag  B ) )
 
Theorembj-2upleq 32505 Substitution property for (|  - ,  - |). (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  ->  ( C  =  D  -> (| A,  C|)  = (| B,  D|) ) )
 
Theorembj-pr21val 32506 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
 |- pr1 (| A,  B|)  =  A
 
Syntaxbj-cpr2 32507 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
 class pr2  A
 
Definitiondf-bj-pr2 32508 Definition of the second projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr2eq 32509, bj-pr22val 32512, bj-pr2ex 32513. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |- pr2  A  =  ( 1o Proj  A )
 
Theorembj-pr2eq 32509 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  =  B  -> pr2  A  = pr2  B )
 
Theorembj-pr2un 32510 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
 |- pr2  ( A  u.  B )  =  (pr2  A  u. pr2  B )
 
Theorembj-pr2val 32511 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
 |- pr2  ( { A }  X. tag  B )  =  if ( A  =  1o ,  B ,  (/) )
 
Theorembj-pr22val 32512 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
 |- pr2 (| A,  B|)  =  B
 
Theorembj-pr2ex 32513 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A  e.  V  -> pr2  A  e.  _V )
 
Theorembj-2uplth 32514 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4566). (Contributed by BJ, 6-Oct-2018.)
 |-  ((| A,  B|)  = (| C,  D|)  <->  ( A  =  C  /\  B  =  D )
 )
 
Theorembj-2uplex 32515 A couple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Oct-2018.)
 |-  ((| A,  B|)  e.  _V  <->  ( A  e.  _V 
 /\  B  e.  _V ) )
 
Theorembj-2upln0 32516 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
 |- (| A,  B|)  =/=  (/)
 
Theorembj-2upln1upl 32517 A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have (| A ,  (/)|)  = (| A|). Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 32502 and bj-2upln0 32516 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)
 |- (| A,  B|)  =/= (| C|)
 
21.29.5.13  Set theory: miscellaneous
 
Theorembj-vtoclgfALT 32518 Alternate proof of vtoclgf 3028. Proof from vtoclgft 3020. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/_ x A   &    |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theorembj-pwcfsdom 32519 Remove hypothesis from pwcfsdom 8747. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 8747.) (Contributed by BJ, 14-Sep-2019.)
 |-  ( aleph `  A )  ~<  ( (
 aleph `  A )  ^m  ( cf `  ( aleph `  A ) ) )
 
Theorembj-grur1 32520 Remove hypothesis from grur1a 8986. Illustration of how to remove a "definitional hypothesis". This makes its uses longer, but the theorem feels more self-contained. Looks preferable when the defined term appears only once in the conclusion. (Contributed by BJ, 14-Sep-2019.)
 |-  (
 ( U  e.  Univ  /\  U  e.  U. ( R1 " On ) ) 
 ->  U  =  ( R1
 `  ( U  i^i  On ) ) )
 
21.29.6  Extended real and complex numbers, real and complex projectives lines

In this section, we indroduce several supersets of the set  RR of real numbers and the set  CC of complex numbers.

Once they are given their usual topologies, which are locally compact, both topological spaces have a one-point compactification. They are denoted by RRhat and CChat respectively, defined in df-bj-cchat 32556 and df-bj-rrhat 32558, and the point at infinity is denoted by infty, defined in df-bj-infty 32554.

Both  RR and  CC also have "directional compactifications", denoted respectively by RRbar, defined in df-bj-rrbar 32552 (already defined as  RR*, see df-xr 9422) and CCbar, defined in df-bj-ccbar 32539.

Since CCbar does not seem to be standard, we describe it in some detail. It is obtained by adding to  CC a "point at infinity at the end of each ray with origin at 0". Although CCbar is not an important object in itself, the motivation for introducing it is to provide a common superset to both RRbar and  CC and to define algebraic operations (addition, opposite, multiplication, inverse) as widely as reasonably possible.

Mathematically, CCbar is the quotient of  ( ( CC  X. RR>=0 )  \  { <. 0 ,  0 >. } ) by the diagonal multiplicative action of  RR+ (think of the closed "northern hemisphere" in R^3 identified with  ( CC  X.  RR ), that each open ray from 0 included in the closed northern half-space intersects exactly once).

Since in set.mm, we want to have a genuine inclusion  CC  C_ CCbar, we instead define CCbar as the (disjoint) union of  CC with a circle at infinity denoted by CCinfty. To have a genuine inclusion RRbar  C_ CCbar, we define pinfty and minfty as certain points in CCinfty.

Thanks to this framework, we have the genuine inclusions  RR  C_ RRbar and  RR  C_ RRhat and similarly with the complex systems, and furthermore  RR  C_  CC and similarly for the "overlined" and "hatted" versions.

Furthermore, we define the main algebraic operations on  (CCbar  u. CChat ), which is not very mathematical, but "overloads" the operations, so that one can use the same notation in all cases.

 
21.29.6.1  Diagonal in a Cartesian square

Complements on the idendity relation and definition of the diagonal in the Cartesian square of a set.

 
Theorembj-elid 32521 Characterization of the elements of 
_I. (Contributed by BJ, 22-Jun-2019.)
 |-  ( A  e.  ( V  X.  V )  ->  ( A  e.  _I  <->  ( 1st `  A )  =  ( 2nd `  A ) ) )
 
Theorembj-elopg 32522 Characterization of the elements of an ordered pair (closed form of elop 4558, which actually has one unnecessary hypothesis). (Contributed by BJ, 22-Jun-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( C  e.  <. A ,  B >.  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) ) )
 
Syntaxcdiag2 32523 Syntax for the diagonal of the Cartesian square of a set.
 class Diag
 
Definitiondf-bj-diag 32524 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
 |- Diag  =  ( x  e.  _V  |->  (  _I  i^i  ( x  X.  x ) ) )
 
Theorembj-diagval 32525 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
 |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )
 
Theorembj-eldiag 32526 Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
 |-  ( A  e.  V  ->  ( B  e.  (Diag `  A )  <->  ( B  e.  ( A  X.  A ) 
 /\  ( 1st `  B )  =  ( 2nd `  B ) ) ) )
 
Theorembj-flbi3 32527 The floor of a real number in  [ 0 ,  1 ) is 0. Remark: may shorten the proof of modid 11732 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.)
 |-  ( A  e.  ( 0 [,) 1 )  ->  ( |_ `  A )  =  0 )
 
Theorembj-pirp 32528 Duplication of pirp 35613. (Contributed by Jim Kingdon, 19-Feb-2019.)
 |-  pi  e.  RR+
 
21.29.6.2  Extended numbers and projective lines as sets

TODO(?): replace inftyexpi with a function inftyexpi2pi defined on  ( 0 [,) 1 ) since we plan to put this section as early as possible, before the definition of  pi.

It looks like to define the sets, the addition and the opposite, one only needs some basic results about addition, opposite and ordering, which could use df-plr 9228, df-ltr 9230, df-0r 9231, df-1r 9232, df-ltr 9230. The idea is then to define the order relation directly on RRbar, skipping  RR.

 
Syntaxcinftyexpi 32529 Syntax for the function inftyexpi parameterizing CCinfty.
 class inftyexpi
 
Definitiondf-bj-inftyexpi 32530 Definition of the auxiliary function inftyexpi parameterizing the circle at infinity CCinfty in CCbar. We use coupling with  CC to simplify the proof of bj-ccinftydisj 32536. It could seem more natural to define inftyexpi on all of  RR using prcpal but we want to use only basic functions in the definition of CCbar. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
 |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <. x ,  CC >. )
 
Theorembj-inftyexpiinv 32531 Utility theorem for the inverse of inftyexpi. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
 |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  A )
 
Theorembj-inftyexpiinj 32532 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32531 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 32533 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 32534 Syntax for the circle at infinity CCinfty.
 class CCinfty
 
Definitiondf-bj-ccinfty 32535 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 32536 The circle at infinity is disjoint form the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
 |-  ( CC  i^i CCinfty )  =  (/)
 
Theorembj-elccinfty 32537 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  e. CCinfty )
 
Syntaxcccbar 32538 Syntax for the set of extended complex numbers CCbar.
 class CCbar
 
Definitiondf-bj-ccbar 32539 Definition of the set of extended complex numbers CCbar. (Contributed by BJ, 22-Jun-2019.)
 |- CCbar  =  ( CC  u. CCinfty )
 
Theorembj-ccssccbar 32540 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |-  CC  C_ CCbar
 
Theorembj-ccinftyssccbar 32541 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
 |- CCinfty  C_ CCbar
 
Syntaxcpinfty 32542 Syntax for pinfty.
 class pinfty
 
Definitiondf-bj-pinfty 32543 Definition of pinfty. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  =  (inftyexpi  `  0 )
 
Theorembj-pinftyccb 32544 The class pinfty is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  e. CCbar
 
Theorembj-pinftynrr 32545 The extended complex number pinfty is not a complex number. (Contributed by BJ, 27-Jun-2019.)
 |-  -. pinfty  e. 
 CC
 
Syntaxcminfty 32546 Syntax for minfty.
 class minfty
 
Definitiondf-bj-minfty 32547 Definition of minfty. (Contributed by BJ, 27-Jun-2019.)
 |- minfty  =  (inftyexpi  `  pi )
 
Theorembj-minftyccb 32548 The class minfty is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
 |- minfty  e. CCbar
 
Theorembj-minftynrr 32549 The extended complex number minfty is not a complex number. (Contributed by BJ, 27-Jun-2019.)
 |-  -. minfty  e. 
 CC
 
Theorembj-pinftynminfty 32550 The extended complex numbers pinfty and minfty are different. (Contributed by BJ, 27-Jun-2019.)
 |- pinfty  =/= minfty
 
Syntaxcrrbar 32551 Syntax for the set of extended real numbers RRbar.
 class RRbar
 
Definitiondf-bj-rrbar 32552 Definition of the set of extended real numbers RRbar. See df-xr 9422. (Contributed by BJ, 29-Jun-2019.)
 |- RRbar  =  ( RR  u.  {minfty , pinfty } )
 
Syntaxcinfty 32553 Syntax for infty.
 class infty
 
Definitiondf-bj-infty 32554 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 32555 Syntax for CChat.
 class CChat
 
Definitiondf-bj-cchat 32556 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
 |- CChat  =  ( CC  u.  {infty } )
 
Syntaxcrrhat 32557 Syntax for RRhat.
 class RRhat
 
Definitiondf-bj-rrhat 32558 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
 |- RRhat  =  ( RR  u.  {infty } )
 
Theorembj-rrhatsscchat 32559 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 32560 Syntax for the addition of extended complex numbers.
 class +cc
 
Definitiondf-bj-addc 32561 Define the additions of extended complex numbers (on the subset of  (CCbar  X. CCbar ) where it makes sense) and of the projective line. (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 32562 Syntax for the opposite of extended complex numbers.
 class -cc
 
Definitiondf-bj-oppc 32563 Define the opposite of an extended complex number or an element of the projective line (or more precisely, the operation giving the opposite, sometimes called negation). 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 32564 Syntax for the function prcpal.
 class prcpal
 
Definitiondf-bj-prcpal 32565 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 32566 Syntax for the argument of a nonzero extended complex number.
 class Arg
 
Definitiondf-bj-arg 32567 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 32568 Syntax for the multiplication of extended complex numbers.
 class .cc
 
Definitiondf-bj-mulc 32569 Define the multiplications of extended complex numbers and of the complex projective line. In our convention, a product with 0 is 0. Another convention leaves products of 0 with an infinite number undefined since the multiplication is not continuous at these points. (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 32570 Syntax for the inverse of nonzero extended complex numbers.
 class invc
 
Definitiondf-bj-invc 32571 Define inversion, which maps a nonzero extended complex number or element of the projective line to its inverse. Beware of the overloading: the equality  (invc `  0
)  = infty is to be understood in the 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 16277 and subsequents. The first few statements of this subsection can be put very early after ccmn 16277. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabel 16278 to "cabl" or other labels containing "abl" to "abel".

 
Theorembj-cmnssmnd 32572 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |- CMnd  C_  Mnd
 
Theorembj-cmnssmndel 32573 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 16292, which relies on iscmn 16284. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e. CMnd  ->  A  e.  Mnd )
 
Theorembj-ablssgrp 32574 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  Abel  C_ 
 Grp
 
Theorembj-ablssgrpel 32575 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 16282. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  Abel  ->  A  e.  Grp )
 
Theorembj-ablsscmn 32576 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  Abel  C_ CMnd
 
Theorembj-ablsscmnel 32577 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 16283. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  Abel  ->  A  e. CMnd )
 
Theorembj-modssabl 32578 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 16992; see also lmodgrp 16955 and lmodcmn 16993.) (Contributed by BJ, 9-Jun-2019.)
 |-  LMod  C_ 
 Abel
 
Theorembj-vecssmod 32579 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
 |-  LVec  C_ 
 LMod
 
Theorembj-vecssmodel 32580 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 17187. (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 14381 (although it mixes finite and infinite sums, which makes it harder to understand).

 
Syntaxcfinsum 32581 Syntax for the class "finite summation in monoids".
 class FinSum
 
Definitiondf-bj-finsum 32582* 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 32583* 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 32584 Syntax for the class of real vector spaces.
 class RR-Vec
 
Definitiondf-bj-rrvec 32585 Definition of the class of real vector spaces. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  =  { x  e.  LVec  |  (Scalar `  x )  = RRfld }
 
Theorembj-rrvecssvec 32586 Real vector spaces are vector spaces. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  C_  LVec
 
Theorembj-rrvecssvecel 32587 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 9-Jun-2019.)
 |-  ( A  e. RR-Vec  ->  A  e.  LVec
 )
 
Theorembj-rrvecsscmn 32588 (The additive groups of) real vector spaces are commutative monoids. (Contributed by BJ, 9-Jun-2019.)
 |- RR-Vec  C_ CMnd
 
Theorembj-rrvecsscmnel 32589 (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 32590 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 32591 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 32592 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 32593 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 32594 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 32595 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 32596 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 32597 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 32598 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 32601 (which computes them).

 
Theorembj-bary1lem 32599 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 32600 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 ) ) ) )
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