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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem00sr 8601 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 8602 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 8603 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 8604* 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 8605* 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 8606 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 8607 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 8608 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 8609* 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 8610 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 8611 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 8612* 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 8613* 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 8614* 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 8615 Class of complex numbers.
 class  CC
 
Syntaxcr 8616 Class of real numbers.
 class  RR
 
Syntaxcc0 8617 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 8618 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 8619 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 8620 Addition on complex numbers.
 class  +
 
Syntaxcltrr 8621 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 8622 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 8623 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8650. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 8624 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 8625 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 8626 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 8627 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-plus 8628* 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 8629* 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 8630* Define 'less than' on the real subset of complex numbers. (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 8631 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 8632 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 8633* 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 8634 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 8635 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 8636 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 8637 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 8638 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 8639 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 8640 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 8641 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 8642 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 8643 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 8644 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6611, 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 8623), 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 8645 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8644 and mulcnsrec 8646. (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 8646 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6610, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8644.

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 8346. (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 8647 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 8653. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8696. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 8648 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 8655. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8697. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxcnex 8649 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10229), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4028 in later theorems by invoking the axiom ax-cnex 8673 instead of cnexALT 10229. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 8650 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 8674. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 8651 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 8675. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremaxicn 8652  _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 8676. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 8653 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; instead, use ax-addcl 8677. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 8654 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; instead, use ax-addrcl 8678. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 8655 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; instead, use ax-mulcl 8679. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 8656 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; instead, use ax-mulrcl 8680. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxmulcom 8657 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; instead, use ax-mulcom 8681. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 8658 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; instead, use ax-addass 8682. (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 8659 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 8683. (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 8660 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; instead, use ax-distr 8684. (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 8661 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 8685. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax1ne0 8662 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 8686. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
 |-  1  =/=  0
 
Theoremax1rid 8663  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 8714, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8687. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremaxrnegex 8664* 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 8688. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxrrecex 8665* 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 8689. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Theoremaxcnre 8666* 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 8690. (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 8667 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 8774. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8691. (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 8668 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 8775. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8692. (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 8669 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 8776. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8693. (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 8670 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 8777. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8694. (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 8671* 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 8778. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8695. (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 8672 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 8673 The complex numbers form a set. This axiom is redundant - see cnexALT 10229- but we provide this axiom because the justification theorem axcnex 8649 does not use ax-rep 4028 even though the redundancy proof does. (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 8674 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8650. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 8675 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8651. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-icn 8676  _i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8652. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 8677 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8653. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Axiomax-addrcl 8678 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8654. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 8679 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8655. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 8680 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8656. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-mulcom 8681 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8657. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Axiomax-addass 8682 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8658. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Axiomax-mulass 8683 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8659. (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 8684 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8660. (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 8685 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 8661. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-1ne0 8686 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 8662. (Contributed by NM, 29-Jan-1995.)
 |-  1  =/=  0
 
Axiomax-1rid 8687  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 8663. Weakened from the original axiom in the form of statement in mulid1 8714, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-rnegex 8688* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 8664. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-rrecex 8689* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8665. (Contributed by Eric Schmidt, 11-Apr-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Axiomax-cnre 8690* 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 8666. (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 8691 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 8667. Note: The more general version for extended reals is axlttri 8774. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Axiomax-pre-lttrn 8692 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 8668. Note: The more general version for extended reals is axlttrn 8775. (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 8693 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 8669. Note: The more general version for extended reals is axltadd 8776. (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 8694 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 8670. Note: The more general version for extended reals is axmulgt0 8777. (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 8695* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 8671. Note: The more general version with ordering on extended reals is axsup 8778. (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 8696 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 ax-addcl 8677 should be used. Note that uses of ax-addf 8696 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 8647. (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 8697 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 8679 should be used. Note that uses of ax-mulf 8697 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 8648. (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 8698 Alias for ax-cnex 8673. See also cnexALT 10229. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 8699 Alias for ax-addcl 8677, for naming consistency with addcli 8721. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 8700 Alias for ax-addrcl 8678, for naming consistency with readdcli 8730. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
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