P. 90 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	enrex 8901	The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
|- ~R e. _V
Theorem	ltrelsr 8902	Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
|- <R C_ ( R. X. R. )
Theorem	addcmpblnr 8903	Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( A +P. F ) , ( B +P. G ) >. ~R <. ( C +P. R ) , ( D +P. S ) >. ) )
Theorem	mulcmpblnrlem 8904	Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
|- ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> ( ( D .P. F ) +P. ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) ) = ( ( D .P. F ) +P. ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
Theorem	mulcmpblnr 8905	Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) )
Theorem	addsrpr 8906	Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
Theorem	mulsrpr 8907	Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
Theorem	ltsrpr 8908	Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )
Theorem	gt0srpr 8909	Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
|- ( 0R <R [ <. A , B >. ] ~R <-> B <P A )
Theorem	0nsr 8910	The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
|- -. (/) e. R.
Theorem	0r 8911	The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- 0R e. R.
Theorem	1sr 8912	The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- 1R e. R.
Theorem	m1r 8913	The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- -1R e. R.
Theorem	addclsr 8914	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. )
Theorem	mulclsr 8915	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. )
Theorem	dmaddsr 8916	Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
|- dom +R = ( R. X. R. )
Theorem	dmmulsr 8917	Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
|- dom .R = ( R. X. R. )
Theorem	addcomsr 8918	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 )
Theorem	addasssr 8919	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 ) )
Theorem	mulcomsr 8920	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 )
Theorem	mulasssr 8921	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 ) )
Theorem	distrsr 8922	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 ) )
Theorem	m1p1sr 8923	Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( -1R +R 1R ) = 0R
Theorem	m1m1sr 8924	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
Theorem	ltsosr 8925	Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
|- <R Or R.
Theorem	0lt1sr 8926	0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
|- 0R <R 1R
Theorem	1ne0sr 8927	1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
|- -. 1R = 0R
Theorem	0idsr 8928	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 )
Theorem	1idsr 8929	1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
|- ( A e. R. -> ( A .R 1R ) = A )
Theorem	00sr 8930	A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
|- ( A e. R. -> ( A .R 0R ) = 0R )
Theorem	ltasr 8931	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 ) ) )
Theorem	pn0sr 8932	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 )
Theorem	negexsr 8933*	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 )
Theorem	recexsrlem 8934*	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 )
Theorem	addgt0sr 8935	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 ) )
Theorem	mulgt0sr 8936	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 ) )
Theorem	sqgt0sr 8937	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 ) )
Theorem	recexsr 8938*	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 )
Theorem	mappsrpr 8939	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. )
Theorem	ltpsrpr 8940	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 )
Theorem	map2psrpr 8941*	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 )
Theorem	supsrlem 8942*	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 ) ) )
Theorem	supsr 8943*	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 ) ) )
Syntax	cc 8944	Class of complex numbers.
class CC
Syntax	cr 8945	Class of real numbers.
class RR
Syntax	cc0 8946	Extend class notation to include the complex number 0.
class 0
Syntax	c1 8947	Extend class notation to include the complex number 1.
class 1
Syntax	ci 8948	Extend class notation to include the complex number i.
class _i
Syntax	caddc 8949	Addition on complex numbers.
class +
Syntax	cltrr 8950	'Less than' predicate (defined over real subset of complex numbers).
class <RR
Syntax	cmul 8951	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 8952	Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8979. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- CC = ( R. X. R. )
Definition	df-0 8953	Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- 0 = <. 0R , 0R >.
Definition	df-1 8954	Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- 1 = <. 1R , 0R >.
Definition	df-i 8955	Define the complex number _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- _i = <. 0R , 1R >.
Definition	df-r 8956	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- RR = ( R. X. { 0R } )
Definition	df-add 8957*	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 ) >. ) ) }
Definition	df-mul 8958*	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 ) ) >. ) ) }
Definition	df-lt 8959*	Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 9081. (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 ) ) }
Theorem	opelcn 8960	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. ) )
Theorem	opelreal 8961	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. )
Theorem	elreal 8962*	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 )
Theorem	elreal2 8963	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 >. ) )
Theorem	0ncn 8964	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
Theorem	ltrelre 8965	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 8966	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 ) >. )
Theorem	mulcnsr 8967	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 ) ) >. )
Theorem	eqresr 8968	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 )
Theorem	addresr 8969	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 >. )
Theorem	mulresr 8970	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 >. )
Theorem	ltresr 8971	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 )
Theorem	ltresr2 8972	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 ) ) )
Theorem	dfcnqs 8973	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6929, 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 8952), 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 )
Theorem	addcnsrec 8974	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8973 and mulcnsrec 8975. (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 )
Theorem	mulcnsrec 8975	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6928, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8973. 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 8675. (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
Theorem	axaddf 8976	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 8982. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 9025. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- + : ( CC X. CC ) --> CC
Theorem	axmulf 8977	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 8984. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9026. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- x. : ( CC X. CC ) --> CC
Theorem	axcnex 8978	The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10564), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4280 in later theorems by invoking the axiom ax-cnex 9002 instead of cnexALT 10564. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 8979	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 9003. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 8980	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 9004. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	axicn 8981	_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 9005. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 8982	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 9006 be used later. Instead, in most cases use addcl 9028. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 8983	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 9007 be used later. Instead, in most cases use readdcl 9029. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 8984	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 9008 be used later. Instead, in most cases use mulcl 9030. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 8985	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 9009 be used later. Instead, in most cases use remulcl 9031. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	axmulcom 8986	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 9010 be used later. Instead, use mulcom 9032. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	axaddass 8987	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 9011 be used later. Instead, use addass 9033. (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 ) ) )
Theorem	axmulass 8988	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 9012. (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 ) ) )
Theorem	axdistr 8989	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 9013 be used later. Instead, use adddi 9035. (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 ) ) )
Theorem	axi2m1 8990	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 9014. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( ( _i x. _i ) + 1 ) = 0
Theorem	ax1ne0 8991	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 9015. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
|- 1 =/= 0
Theorem	ax1rid 8992	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 9044, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9016. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
|- ( A e. RR -> ( A x. 1 ) = A )
Theorem	axrnegex 8993*	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 9017. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Theorem	axrrecex 8994*	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 9018. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ A =/= 0 ) -> E. x e. RR ( A x. x ) = 1 )
Theorem	axcnre 8995*	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 9019. (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 ) ) )
Theorem	axpre-lttri 8996	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 9103. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9020. (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 ) ) )
Theorem	axpre-lttrn 8997	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 9104. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9021. (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 ) )
Theorem	axpre-ltadd 8998	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 9105. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9022. (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 ) ) )
Theorem	axpre-mulgt0 8999	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 9106. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9023. (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 ) ) )
Theorem	axpre-sup 9000*	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 9107. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9024. (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 ) ) )