HomeHome Metamath Proof Explorer
Theorem List (p. 97 of 406)
< 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
(1-26571)
  Hilbert Space Explorer  Hilbert Space Explorer
(26572-28094)
  Users' Mathboxes  Users' Mathboxes
(28095-40593)
 

Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-mulcl 9601 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 9577. Proofs should normally use mulcl 9623 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 9602 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 9578. Proofs should normally use remulcl 9624 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-mulcom 9603 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 9579. Proofs should normally use mulcom 9625 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Axiomax-addass 9604 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 9580. Proofs should normally use addass 9626 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Axiomax-mulass 9605 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9581. Proofs should normally use mulass 9627 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Axiomax-distr 9606 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9582. Proofs should normally use adddi 9628 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Axiomax-i2m1 9607 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 9583. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-1ne0 9608 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 9584. (Contributed by NM, 29-Jan-1995.)
 |-  1  =/=  0
 
Axiomax-1rid 9609  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 9585. Weakened from the original axiom in the form of statement in mulid1 9640, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-rnegex 9610* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 9586. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-rrecex 9611* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9587. (Contributed by Eric Schmidt, 11-Apr-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Axiomax-cnre 9612* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 9588. For naming consistency, use cnre 9639 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Axiomax-pre-lttri 9613 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9589. Note: The more general version for extended reals is axlttri 9705. Normally new proofs would use xrlttri 11438. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Axiomax-pre-lttrn 9614 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9590. Note: The more general version for extended reals is axlttrn 9706. Normally new proofs would use lttr 9710. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <RR  B 
 /\  B  <RR  C ) 
 ->  A  <RR  C ) )
 
Axiomax-pre-ltadd 9615 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9591. Normally new proofs would use axltadd 9707. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A ) 
 <RR  ( C  +  B ) ) )
 
Axiomax-pre-mulgt0 9616 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9592. Normally new proofs would use axmulgt0 9708. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
 
Axiomax-pre-sup 9617* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 9593. Note: Normally new proofs would use axsup 9709. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x ) 
 ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
 y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
 
Axiomax-addf 9618 Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 9621 should be used. Note that uses of ax-addf 9618 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( x  +  y ) ) in place of  +, from which this axiom (with the defined operation in place of  +) follows as a theorem.

This axiom is justified by theorem axaddf 9569. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 9619 Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 9601 should be used. Note that uses of ax-mulf 9619 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( x  x.  y ) ) in place of  x., from which this axiom (with the defined operation in place of  x.) follows as a theorem.

This axiom is justified by theorem axmulf 9570. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
5.2  Derive the basic properties from the field axioms
 
5.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 9620 Alias for ax-cnex 9595. See also cnexALT 11298. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 9621 Alias for ax-addcl 9599, for naming consistency with addcli 9647. Use this theorem instead of ax-addcl 9599 or axaddcl 9575. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 9622 Alias for ax-addrcl 9600, for naming consistency with readdcli 9656. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremmulcl 9623 Alias for ax-mulcl 9601, for naming consistency with mulcli 9648. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremremulcl 9624 Alias for ax-mulrcl 9602, for naming consistency with remulcli 9657. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremmulcom 9625 Alias for ax-mulcom 9603, for naming consistency with mulcomi 9649. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddass 9626 Alias for ax-addass 9604, for naming consistency with addassi 9651. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremmulass 9627 Alias for ax-mulass 9605, for naming consistency with mulassi 9652. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremadddi 9628 Alias for ax-distr 9606, for naming consistency with adddii 9653. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremrecn 9629 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  RR  ->  A  e.  CC )
 
Theoremreex 9630 The real numbers form a set. See also reexALT 11296. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 RR  e.  _V
 
Theoremreelprrecn 9631 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- 
 RR  e.  { RR ,  CC }
 
Theoremcnelprrecn 9632 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- 
 CC  e.  { RR ,  CC }
 
Theoremelimne0 9633 Hypothesis for weak deduction theorem to eliminate  A  =/=  0. (Contributed by NM, 15-May-1999.)
 |- 
 if ( A  =/=  0 ,  A , 
 1 )  =/=  0
 
Theoremadddir 9634 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
 
Theorem0cn 9635 0 is a complex number. See also 0cnALT 9864. (Contributed by NM, 19-Feb-2005.)
 |-  0  e.  CC
 
Theorem0cnd 9636 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  CC )
 
Theoremc0ex 9637 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  e.  _V
 
Theorem1ex 9638 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  1  e.  _V
 
Theoremcnre 9639* Alias for ax-cnre 9612, for naming consistency. (Contributed by NM, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremmulid1 9640  1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  1
 )  =  A )
 
Theoremmulid2 9641 Identity law for multiplication. Note: see mulid1 9640 for commuted version. (Contributed by NM, 8-Oct-1999.)
 |-  ( A  e.  CC  ->  ( 1  x.  A )  =  A )
 
Theorem1re 9642  1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 9597, by exploiting properties of the imaginary unit  _i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  1  e.  RR
 
Theorem0re 9643  0 is a real number. See also 0reALT 9972. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  0  e.  RR
 
Theorem0red 9644  0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  0  e.  RR )
 
Theoremmulid1i 9645 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( A  x.  1 )  =  A
 
Theoremmulid2i 9646 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( 1  x.  A )  =  A
 
Theoremaddcli 9647 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  e.  CC
 
Theoremmulcli 9648 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  e.  CC
 
Theoremmulcomi 9649 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  =  ( B  x.  A )
 
Theoremmulcomli 9650 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  x.  B )  =  C   =>    |-  ( B  x.  A )  =  C
 
Theoremaddassi 9651 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  +  C )  =  ( A  +  ( B  +  C )
 )
 
Theoremmulassi 9652 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) )
 
Theoremadddii 9653 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )
 
Theoremadddiri 9654 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
 
Theoremrecni 9655 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
 |-  A  e.  RR   =>    |-  A  e.  CC
 
Theoremreaddcli 9656 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  +  B )  e.  RR
 
Theoremremulcli 9657 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  x.  B )  e.  RR
 
Theorem1red 9658 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  RR )
 
Theorem1cnd 9659 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  CC )
 
Theoremmulid1d 9660 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  1 )  =  A )
 
Theoremmulid2d 9661 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  x.  A )  =  A )
 
Theoremaddcld 9662 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  e.  CC )
 
Theoremmulcld 9663 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  B )  e.  CC )
 
Theoremmulcomd 9664 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddassd 9665 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  C )  =  ( A  +  ( B  +  C )
 ) )
 
Theoremmulassd 9666 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremadddid 9667 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremadddird 9668 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
 
Theoremrecnd 9669 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremreaddcld 9670 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  +  B )  e.  RR )
 
Theoremremulcld 9671 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  x.  B )  e.  RR )
 
5.2.2  Infinity and the extended real number system
 
Syntaxcpnf 9672 Plus infinity.
 class +oo
 
Syntaxcmnf 9673 Minus infinity.
 class -oo
 
Syntaxcxr 9674 The set of extended reals (includes plus and minus infinity).
 class  RR*
 
Syntaxclt 9675 'Less than' predicate (extended to include the extended reals).
 class  <
 
Syntaxcle 9676 Extend wff notation to include the 'less than or equal to' relation.
 class  <_
 
Definitiondf-pnf 9677 Define plus infinity. Note that the definition is arbitrary, requiring only that +oo be a set not in  RR and different from -oo (df-mnf 9678). We use  ~P
U. CC to make it independent of the construction of  CC, and Cantor's Theorem will show that it is different from any member of 
CC and therefore  RR. See pnfnre 9682, mnfnre 9683, and pnfnemnf 11417.

A simpler possibility is to define +oo as  CC and -oo as  { CC }, but that approach requires the Axiom of Regularity to show that +oo and -oo are different from each other and from all members of  RR. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

 |- +oo  =  ~P U. CC
 
Definitiondf-mnf 9678 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -oo be a set not in  RR and different from +oo (see mnfnre 9683 and pnfnemnf 11417). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
 |- -oo  =  ~P +oo
 
Definitiondf-xr 9679 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)
 |-  RR*  =  ( RR  u.  { +oo , -oo } )
 
Definitiondf-ltxr 9680* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers,  <RR is primitive and not necessarily a relation on  RR. (Contributed by NM, 13-Oct-2005.)
 |- 
 <  =  ( { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  ( ( ( RR 
 u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) )
 
Definitiondf-le 9681 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 9720 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
 |- 
 <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
 
Theorempnfnre 9682 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- +oo  e/  RR
 
Theoremmnfnre 9683 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- -oo  e/  RR
 
Theoremressxr 9684 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
 |- 
 RR  C_  RR*
 
Theoremrexpssxrxp 9685 The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
 
Theoremrexr 9686 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  e.  RR* )
 
Theorem0xr 9687 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  0  e.  RR*
 
Theoremrenepnf 9688 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( A  e.  RR  ->  A  =/= +oo )
 
Theoremrenemnf 9689 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( A  e.  RR  ->  A  =/= -oo )
 
Theoremrexrd 9690 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrenepnfd 9691 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremrenemnfd 9692 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= -oo )
 
Theoremrexri 9693 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  A  e.  RR*
 
Theoremrenfdisj 9694 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( RR  i^i  { +oo , -oo } )  =  (/)
 
Theoremltrelxr 9695 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <  C_  ( RR*  X.  RR* )
 
Theoremltrel 9696 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
 |- 
 Rel  <
 
Theoremlerelxr 9697 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <_  C_  ( RR*  X.  RR* )
 
Theoremlerel 9698 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  <_
 
Theoremxrlenlt 9699 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
 
Theoremxrlenltd 9700 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A  <_  B  <->  -.  B  <  A ) )
    < 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 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40593
  Copyright terms: Public domain < Previous  Next >