P. 82 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axarch 8101*	Archimedean axiom. The Archimedean property is more naturally stated once we have defined NN. Unless we find another way to state it, we'll just use the right hand side of dfnn2 9135 in stating what we mean by "natural number" in the context of this axiom. This construction-dependent theorem should not be referenced directly; instead, use ax-arch 8141. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)
|- ( A e. RR -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )
Theorem	peano5nnnn 8102*	Peano's inductive postulate. This is a counterpart to peano5nni 9136 designed for real number axioms which involve natural numbers (notably, axcaucvg 8110). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A )
Theorem	nnindnn 8103*	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 9149 designed for real number axioms which involve natural numbers (notably, axcaucvg 8110). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( z = 1 -> ( ph <-> ps ) )   &    |- ( z = k -> ( ph <-> ch ) )   &    |- ( z = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( z = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( k e. N -> ( ch -> th ) )   =>    |- ( A e. N -> ta )
Theorem	nntopi 8104*	Mapping from NN to N.. (Contributed by Jim Kingdon, 13-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( A e. N -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A )
Theorem	axcaucvglemcl 8105*	Lemma for axcaucvg 8110. Mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   =>    |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) e. R. )
Theorem	axcaucvglemf 8106*	Lemma for axcaucvg 8110. Mapping to N. and R. yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> G : N. --> R. )
Theorem	axcaucvglemval 8107*	Lemma for axcaucvg 8110. Value of sequence when mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. )
Theorem	axcaucvglemcau 8108*	Lemma for axcaucvg 8110. The result of mapping to N. and R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
Theorem	axcaucvglemres 8109*	Lemma for axcaucvg 8110. Mapping the limit from N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
Theorem	axcaucvg 8110*	Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). Because we are stating this axiom before we have introduced notations for NN or division, we use N for the natural numbers and express a reciprocal in terms of iota_. This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 8142. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
Theorem	axpre-suploclemres 8111*	Lemma for axpre-suploc 8112. The result. The proof just needs to define B as basically the same set as A (but expressed as a subset of R. rather than a subset of RR), and apply suplocsr 8019. (Contributed by Jim Kingdon, 24-Jan-2024.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. RR A. y e. A y <RR x )   &    |- ( ph -> A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) )   &    |- B = { w e. R. | <. w , 0R >. e. A }   =>    |- ( ph -> 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	axpre-suploc 8112*	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given x < y, either there is an element of the set greater than x, or y is an upper bound. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 8143. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y <RR x /\ A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) ) -> 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 ) ) )
4.1.3  Real and complex number postulates restated as axioms
Axiom	ax-cnex 8113	The complex numbers form a set. Proofs should normally use cnex 8146 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
|- CC e. _V
Axiom	ax-resscn 8114	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 8070. (Contributed by NM, 1-Mar-1995.)
|- RR C_ CC
Axiom	ax-1cn 8115	1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 8071. (Contributed by NM, 1-Mar-1995.)
|- 1 e. CC
Axiom	ax-1re 8116	1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 8072. Proofs should use 1re 8168 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
|- 1 e. RR
Axiom	ax-icn 8117	_i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 8073. (Contributed by NM, 1-Mar-1995.)
|- _i e. CC
Axiom	ax-addcl 8118	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 8074. Proofs should normally use addcl 8147 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 8119	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 8075. Proofs should normally use readdcl 8148 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Axiom	ax-mulcl 8120	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 8076. Proofs should normally use mulcl 8149 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 8121	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 8077. Proofs should normally use remulcl 8150 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Axiom	ax-addcom 8122	Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 8080. Proofs should normally use addcom 8306 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Axiom	ax-mulcom 8123	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8081. Proofs should normally use mulcom 8151 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 8124	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8082. Proofs should normally use addass 8152 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 8125	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8083. Proofs should normally use mulass 8153 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 8126	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 8084. Proofs should normally use adddi 8154 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 8127	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 8085. (Contributed by NM, 29-Jan-1995.)
|- ( ( _i x. _i ) + 1 ) = 0
Axiom	ax-0lt1 8128	0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 8086. Proofs should normally use 0lt1 8296 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
|- 0 <RR 1
Axiom	ax-1rid 8129	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 8087. (Contributed by NM, 29-Jan-1995.)
|- ( A e. RR -> ( A x. 1 ) = A )
Axiom	ax-0id 8130	0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 8088. Proofs should normally use addrid 8307 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Axiom	ax-rnegex 8131*	Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 8089. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Axiom	ax-precex 8132*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 8090. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
Axiom	ax-cnre 8133*	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by Theorem axcnre 8091. For naming consistency, use cnre 8165 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-ltirr 8134	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 8134. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( A e. RR -> -. A <RR A )
Axiom	ax-pre-ltwlin 8135	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 8093. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
Axiom	ax-pre-lttrn 8136	Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 8094. (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-apti 8137	Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 8095. (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
Axiom	ax-pre-ltadd 8138	Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 8096. (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 8139	The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 8097. (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-mulext 8140	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, justified by Theorem axpre-mulext 8098 (Contributed by Jim Kingdon, 18-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) )
Axiom	ax-arch 8141*	Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by Theorem axarch 8101. This axiom should not be used directly; instead use arch 9389 (which is the same, but stated in terms of NN and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)
|- ( A e. RR -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )
Axiom	ax-caucvg 8142*	Completeness. Axiom for real and complex numbers, justified by Theorem axcaucvg 8110. A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / n of the nth term. This axiom should not be used directly; instead use caucvgre 11532 (which is the same, but stated in terms of the NN and 1 / n notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
Axiom	ax-pre-suploc 8143*	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given x < y, either there is an element of the set greater than x, or y is an upper bound. Although this and ax-caucvg 8142 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 8142. (Contributed by Jim Kingdon, 23-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y <RR x /\ A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) ) -> 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 8144	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 https://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 8147 should be used. Note that uses of ax-addf 8144 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 8078. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
|- + : ( CC X. CC ) --> CC
Axiom	ax-mulf 8145	Multiplication is an operation on the complex numbers. This axiom tells us that x. is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 8371 or eff 12214. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 8149. Note that uses of ax-mulf 8145 can be eliminated by using the defined operation ( x e. CC , y e. CC |-> ( x x. y ) ) in place of x., as seen in mpomulf 8159. This axiom is justified by Theorem axmulf 8079. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
|- x. : ( CC X. CC ) --> CC
4.2  Derive the basic properties from the field axioms
4.2.1  Some deductions from the field axioms for complex numbers
Theorem	cnex 8146	Alias for ax-cnex 8113. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- CC e. _V
Theorem	addcl 8147	Alias for ax-addcl 8118, for naming consistency with addcli 8173. Use this theorem instead of ax-addcl 8118 or axaddcl 8074. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	readdcl 8148	Alias for ax-addrcl 8119, for naming consistency with readdcli 8182. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	mulcl 8149	Alias for ax-mulcl 8120, for naming consistency with mulcli 8174. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	remulcl 8150	Alias for ax-mulrcl 8121, for naming consistency with remulcli 8183. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	mulcom 8151	Alias for ax-mulcom 8123, for naming consistency with mulcomi 8175. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	addass 8152	Alias for ax-addass 8124, for naming consistency with addassi 8177. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulass 8153	Alias for ax-mulass 8125, for naming consistency with mulassi 8178. (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 8154	Alias for ax-distr 8126, for naming consistency with adddii 8179. (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 8155	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
|- ( A e. RR -> A e. CC )
Theorem	reex 8156	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- RR e. _V
Theorem	reelprrecn 8157	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 8158	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	mpomulf 8159*	Multiplication is an operation on complex numbers. Version of ax-mulf 8145 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8120. (Contributed by GG, 16-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC
Theorem	adddir 8160	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 8161	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
|- 0 e. CC
Theorem	0cnd 8162	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. CC )
Theorem	c0ex 8163	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 e. _V
Theorem	1ex 8164	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 1 e. _V
Theorem	cnre 8165*	Alias for ax-cnre 8133, 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	mulrid 8166	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	mullid 8167	Identity law for multiplication. Note: see mulrid 8166 for commuted version. (Contributed by NM, 8-Oct-1999.)
|- ( A e. CC -> ( 1 x. A ) = A )
Theorem	1re 8168	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
|- 1 e. RR
Theorem	0re 8169	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
|- 0 e. RR
Theorem	0red 8170	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 0 e. RR )
Theorem	mulridi 8171	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( A x. 1 ) = A
Theorem	mullidi 8172	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( 1 x. A ) = A
Theorem	addcli 8173	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) e. CC
Theorem	mulcli 8174	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) e. CC
Theorem	mulcomi 8175	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) = ( B x. A )
Theorem	mulcomli 8176	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 8177	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 8178	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 8179	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 8180	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 8181	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
|- A e. RR   =>    |- A e. CC
Theorem	readdcli 8182	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A + B ) e. RR
Theorem	remulcli 8183	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 8184	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 8185	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulridd 8186	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mullidd 8187	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	mulid2d 8188	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 8189	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 8190	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 8191	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 8192	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 8193	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 8194	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 8195	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	adddirp1d 8196	Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )
Theorem	joinlmuladdmuld 8197	Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D )   =>    |- ( ph -> ( ( A + C ) x. B ) = D )
Theorem	recnd 8198	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 8199	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 8200	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 )