P. 97 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axi2m1 9601	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 9625. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( ( _i x. _i ) + 1 ) = 0
Theorem	ax1ne0 9602	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 9626. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
|- 1 =/= 0
Theorem	ax1rid 9603	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 9658, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9627. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
|- ( A e. RR -> ( A x. 1 ) = A )
Theorem	axrnegex 9604*	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 9628. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Theorem	axrrecex 9605*	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 9629. (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 9606*	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 9630. (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 9607	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 9723. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9631. (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 9608	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 9724. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9632. (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 9609	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 9725. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9633. (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 9610	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 9726. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9634. (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 9611*	A nonempty, 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 9727. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9635. (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 ) ) )
Theorem	wuncn 9612	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
Axiom	ax-cnex 9613	The complex numbers form a set. This axiom is redundant - see cnexALT 11321- but we provide this axiom because the justification theorem axcnex 9589 does not use ax-rep 4508 even though the redundancy proof does. Proofs should normally use cnex 9638 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
|- CC e. _V
Axiom	ax-resscn 9614	The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 9590. (Contributed by NM, 1-Mar-1995.)
|- RR C_ CC
Axiom	ax-1cn 9615	1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 9591. (Contributed by NM, 1-Mar-1995.)
|- 1 e. CC
Axiom	ax-icn 9616	_i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 9592. (Contributed by NM, 1-Mar-1995.)
|- _i e. CC
Axiom	ax-addcl 9617	Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9593. Proofs should normally use addcl 9639 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 )
Axiom	ax-addrcl 9618	Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 9594. Proofs should normally use readdcl 9640 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Axiom	ax-mulcl 9619	Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 9595. Proofs should normally use mulcl 9641 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Axiom	ax-mulrcl 9620	Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 9596. Proofs should normally use remulcl 9642 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Axiom	ax-mulcom 9621	Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 9597. Proofs should normally use mulcom 9643 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Axiom	ax-addass 9622	Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 9598. Proofs should normally use addass 9644 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 ) ) )
Axiom	ax-mulass 9623	Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9599. Proofs should normally use mulass 9645 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 ) ) )
Axiom	ax-distr 9624	Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9600. Proofs should normally use adddi 9646 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 ) ) )
Axiom	ax-i2m1 9625	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 9601. (Contributed by NM, 29-Jan-1995.)
|- ( ( _i x. _i ) + 1 ) = 0
Axiom	ax-1ne0 9626	1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 9602. (Contributed by NM, 29-Jan-1995.)
|- 1 =/= 0
Axiom	ax-1rid 9627	1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 9603. Weakened from the original axiom in the form of statement in mulid1 9658, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
|- ( A e. RR -> ( A x. 1 ) = A )
Axiom	ax-rnegex 9628*	Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 9604. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Axiom	ax-rrecex 9629*	Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9605. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- ( ( A e. RR /\ A =/= 0 ) -> E. x e. RR ( A x. x ) = 1 )
Axiom	ax-cnre 9630*	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 9606. For naming consistency, use cnre 9657 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 ) ) )
Axiom	ax-pre-lttri 9631	Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9607. Note: The more general version for extended reals is axlttri 9723. Normally new proofs would use xrlttri 11461. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> -. ( A = B \/ B <RR A ) ) )
Axiom	ax-pre-lttrn 9632	Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9608. Note: The more general version for extended reals is axlttrn 9724. Normally new proofs would use lttr 9728. (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 ) )
Axiom	ax-pre-ltadd 9633	Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9609. Normally new proofs would use axltadd 9725. (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 ) ) )
Axiom	ax-pre-mulgt0 9634	The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9610. Normally new proofs would use axmulgt0 9726. (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 ) ) )
Axiom	ax-pre-sup 9635*	A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 9611. Note: Normally new proofs would use axsup 9727. (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 ) ) )
Axiom	ax-addf 9636	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 9639 should be used. Note that uses of ax-addf 9636 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 9587. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
|- + : ( CC X. CC ) --> CC
Axiom	ax-mulf 9637	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 9619 should be used. Note that uses of ax-mulf 9637 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 9588. (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
Theorem	cnex 9638	Alias for ax-cnex 9613. See also cnexALT 11321. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- CC e. _V
Theorem	addcl 9639	Alias for ax-addcl 9617, for naming consistency with addcli 9665. Use this theorem instead of ax-addcl 9617 or axaddcl 9593. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	readdcl 9640	Alias for ax-addrcl 9618, for naming consistency with readdcli 9674. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	mulcl 9641	Alias for ax-mulcl 9619, for naming consistency with mulcli 9666. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	remulcl 9642	Alias for ax-mulrcl 9620, for naming consistency with remulcli 9675. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	mulcom 9643	Alias for ax-mulcom 9621, for naming consistency with mulcomi 9667. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	addass 9644	Alias for ax-addass 9622, for naming consistency with addassi 9669. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulass 9645	Alias for ax-mulass 9623, for naming consistency with mulassi 9670. (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 ) ) )
Theorem	adddi 9646	Alias for ax-distr 9624, for naming consistency with adddii 9671. (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 ) ) )
Theorem	recn 9647	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
|- ( A e. RR -> A e. CC )
Theorem	reex 9648	The real numbers form a set. See also reexALT 11319. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- RR e. _V
Theorem	reelprrecn 9649	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 }
Theorem	cnelprrecn 9650	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 }
Theorem	elimne0 9651	Hypothesis for weak deduction theorem to eliminate A =/= 0. (Contributed by NM, 15-May-1999.)
|- if ( A =/= 0 , A , 1 ) =/= 0
Theorem	adddir 9652	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 ) ) )
Theorem	0cn 9653	0 is a complex number. See also 0cnALT 9884. (Contributed by NM, 19-Feb-2005.)
|- 0 e. CC
Theorem	0cnd 9654	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. CC )
Theorem	c0ex 9655	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 e. _V
Theorem	1ex 9656	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 1 e. _V
Theorem	cnre 9657*	Alias for ax-cnre 9630, 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 ) ) )
Theorem	mulid1 9658	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 )
Theorem	mulid2 9659	Identity law for multiplication. Note: see mulid1 9658 for commuted version. (Contributed by NM, 8-Oct-1999.)
|- ( A e. CC -> ( 1 x. A ) = A )
Theorem	1re 9660	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 9615, 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
Theorem	0re 9661	0 is a real number. See also 0reALT 9991. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
|- 0 e. RR
Theorem	0red 9662	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 0 e. RR )
Theorem	mulid1i 9663	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( A x. 1 ) = A
Theorem	mulid2i 9664	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( 1 x. A ) = A
Theorem	addcli 9665	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) e. CC
Theorem	mulcli 9666	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) e. CC
Theorem	mulcomi 9667	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) = ( B x. A )
Theorem	mulcomli 9668	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- ( A x. B ) = C   =>    |- ( B x. A ) = C
Theorem	addassi 9669	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 ) )
Theorem	mulassi 9670	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 ) )
Theorem	adddii 9671	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 ) )
Theorem	adddiri 9672	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 ) )
Theorem	recni 9673	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
|- A e. RR   =>    |- A e. CC
Theorem	readdcli 9674	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A + B ) e. RR
Theorem	remulcli 9675	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A x. B ) e. RR
Theorem	1red 9676	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 9677	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulid1d 9678	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mulid2d 9679	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 9680	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A + B ) e. CC )
Theorem	mulcld 9681	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 )
Theorem	mulcomd 9682	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 ) )
Theorem	addassd 9683	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 ) ) )
Theorem	mulassd 9684	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 ) ) )
Theorem	adddid 9685	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 ) ) )
Theorem	adddird 9686	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 ) ) )
Theorem	recnd 9687	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 9688	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 )
Theorem	remulcld 9689	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
Syntax	cpnf 9690	Plus infinity.
class +oo
Syntax	cmnf 9691	Minus infinity.
class -oo
Syntax	cxr 9692	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 9693	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 9694	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 9695	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 9696). 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 9700, mnfnre 9701, and pnfnemnf 11440. 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
Definition	df-mnf 9696	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 9701 and pnfnemnf 11440). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 9697	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 } )
Definition	df-ltxr 9698*	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 ) ) )
Definition	df-le 9699	Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 9738 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
|- <_ = ( ( RR* X. RR* ) \ `' < )
Theorem	pnfnre 9700	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR