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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddclsr 8701 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  e.  R. )
 
Theoremmulclsr 8702 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  e.  R. )
 
Theoremdmaddsr 8703 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  +R  =  ( R.  X.  R. )
 
Theoremdmmulsr 8704 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  .R  =  ( R. 
 X.  R. )
 
Theoremaddcomsr 8705 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  +R  B )  =  ( B  +R  A )
 
Theoremaddasssr 8706 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )
 
Theoremmulcomsr 8707 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  B )  =  ( B  .R  A )
 
Theoremmulasssr 8708 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) )
 
Theoremdistrsr 8709 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  ( B  +R  C ) )  =  ( ( A 
 .R  B )  +R  ( A  .R  C ) )
 
Theoremm1p1sr 8710 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 8711 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremltsosr 8712 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 8713 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 8714 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 8715 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
 
Theorem1idsr 8716 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 8717 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasr 8718 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
 
Theorempn0sr 8719 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 8720* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  E. x  e.  R.  ( A  +R  x )  =  0R )
 
Theoremrecexsrlem 8721* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremaddgt0sr 8722 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  +R  B ) )
 
Theoremmulgt0sr 8723 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  .R  B ) )
 
Theoremsqgt0sr 8724 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  0R  <R  ( A 
 .R  A ) )
 
Theoremrecexsr 8725* The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremmappsrpr 8726 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  ) 
 <->  A  e.  P. )
 
Theoremltpsrpr 8727 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  [ <. A ,  1P >. ]  ~R  )  <R  ( C  +R  [ <. B ,  1P >. ] 
 ~R  )  <->  A  <P  B )
 
Theoremmap2psrpr 8728* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  A  <->  E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
 
Theoremsupsrlem 8729* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  B  =  { w  |  ( C  +R  [ <. w ,  1P >. ] 
 ~R  )  e.  A }   &    |-  C  e.  R.   =>    |-  ( ( C  e.  A  /\  E. x  e.  R.  A. y  e.  A  y  <R  x ) 
 ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y  /\  A. y  e.  R.  (
 y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Theoremsupsr 8730* A non-empty, bounded set of signed reals has a supremum. (Cotributed by Mario Carneiro, 15-Jun-2013.) (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  R.  A. y  e.  A  y 
 <R  x )  ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y 
 /\  A. y  e.  R.  ( y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Syntaxcc 8731 Class of complex numbers.
 class  CC
 
Syntaxcr 8732 Class of real numbers.
 class  RR
 
Syntaxcc0 8733 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 8734 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 8735 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 8736 Addition on complex numbers.
 class  +
 
Syntaxcltrr 8737 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 8738 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 8739 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8766. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 8740 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 8741 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 8742 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 8743 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-add 8744* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
 |- 
 +  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( w  +R  u ) ,  ( v  +R  f ) >. ) ) }
 
Definitiondf-mul 8745* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 x.  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v 
 .R  u )  +R  ( w  .R  f ) ) >. ) ) }
 
Definitiondf-lt 8746* Define 'less than' on the real subset of complex numbers. Proofs should typically use  < instead; see df-ltxr 8868. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  = 
 <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
 
Theoremopelcn 8747 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( <. A ,  B >.  e.  CC  <->  ( A  e.  R. 
 /\  B  e.  R. ) )
 
Theoremopelreal 8748 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( <. A ,  0R >.  e.  RR  <->  A  e.  R. )
 
Theoremelreal 8749* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  <->  E. x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelreal2 8750 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  <->  (
 ( 1st `  A )  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
 
Theorem0ncn 8751 The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  CC
 
Theoremltrelre 8752 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 8753 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( <. A ,  B >.  +  <. C ,  D >. )  =  <. ( A  +R  C ) ,  ( B  +R  D ) >. )
 
Theoremmulcnsr 8754 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( <. A ,  B >.  x.  <. C ,  D >. )  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  (
 ( B  .R  C )  +R  ( A  .R  D ) ) >. )
 
Theoremeqresr 8755 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( <. A ,  0R >.  =  <. B ,  0R >. 
 <->  A  =  B )
 
Theoremaddresr 8756 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  +  <. B ,  0R >. )  =  <. ( A  +R  B ) ,  0R >. )
 
Theoremmulresr 8757 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  x.  <. B ,  0R >. )  =  <. ( A  .R  B ) ,  0R >. )
 
Theoremltresr 8758 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( <. A ,  0R >.  <RR 
 <. B ,  0R >.  <->  A  <R  B )
 
Theoremltresr2 8759 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <-> 
 ( 1st `  A )  <R  ( 1st `  B ) ) )
 
Theoremdfcnqs 8760 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6721, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that  CC is a quotient set, even though it is not (compare df-c 8739), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
 |- 
 CC  =  ( ( R.  X.  R. ) /. `'  _E  )
 
Theoremaddcnsrec 8761 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8760 and mulcnsrec 8762. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( [ <. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  )
 
Theoremmulcnsrec 8762 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6720, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8760.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 8462. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( [ <. A ,  B >. ] `'  _E  x.  [
 <. C ,  D >. ] `'  _E  )  =  [ <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B 
 .R  C )  +R  ( A  .R  D ) ) >. ] `'  _E  )
 
5.1.2  Final derivation of real and complex number postulates
 
Theoremaxaddf 8763 Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 8769. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8812. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 8764 Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 8771. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8813. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxcnex 8765 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10346), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4132 in later theorems by invoking the axiom ax-cnex 8789 instead of cnexALT 10346. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 8766 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 8790. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 8767 1 is a complex number. Axiom 2 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 8791. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremaxicn 8768  _i is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 8792. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 8769 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 8793 be used later. Instead, in most cases use addcl 8815. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 8770 Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 8794 be used later. Instead, in most cases use readdcl 8816. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 8771 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 8795 be used later. Instead, in most cases use mulcl 8817. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 8772 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 8796 be used later. Instead, in most cases use remulcl 8818. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxmulcom 8773 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 8797 be used later. Instead, use mulcom 8819. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 8774 Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 8798 be used later. Instead, use addass 8820. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremaxmulass 8775 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 8799. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremaxdistr 8776 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 8800 be used later. Instead, use adddi 8822. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremaxi2m1 8777 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 8801. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax1ne0 8778 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 8802. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
 |-  1  =/=  0
 
Theoremax1rid 8779  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 8831, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8803. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremaxrnegex 8780* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 8804. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxrrecex 8781* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 8805. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Theoremaxcnre 8782* 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, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 8806. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremaxpre-lttri 8783 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 8890. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8807. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Theoremaxpre-lttrn 8784 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 8891. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8808. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <RR  B 
 /\  B  <RR  C ) 
 ->  A  <RR  C ) )
 
Theoremaxpre-ltadd 8785 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 8892. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8809. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A ) 
 <RR  ( C  +  B ) ) )
 
Theoremaxpre-mulgt0 8786 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 8893. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8810. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
 
Theoremaxpre-sup 8787* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 8894. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8811. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( 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 ) ) )
 
Theoremwuncn 8788 A weak universe containing  om contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ph  ->  CC  e.  U )
 
5.1.3  Real and complex number postulates restated as axioms
 
Axiomax-cnex 8789 The complex numbers form a set. This axiom is redundant - see cnexALT 10346- but we provide this axiom because the justification theorem axcnex 8765 does not use ax-rep 4132 even though the redundancy proof does. Proofs should normally use cnex 8814 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 8790 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8766. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 8791 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8767. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-icn 8792  _i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8768. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 8793 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8769. Proofs should normally use addcl 8815 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Axiomax-addrcl 8794 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8770. Proofs should normally use readdcl 8816 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 8795 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8771. Proofs should normally use mulcl 8817 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 8796 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8772. Proofs should normally use remulcl 8818 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-mulcom 8797 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8773. Proofs should normally use mulcom 8819 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 8798 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8774. Proofs should normally use addass 8820 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 8799 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8775. Proofs should normally use mulass 8821 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 8800 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8776. Proofs should normally use adddi 8822 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 ) ) )
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