P. 83 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axpre-ltadd 8201	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8243. (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 8202	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8244. (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-mulext 8203	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 8245. (Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) )
Theorem	rereceu 8204*	The reciprocal from axprecex 8195 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( ( A e. RR /\ 0 <RR A ) -> E! x e. RR ( A x. x ) = 1 )
Theorem	recriota 8205*	Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( N e. N. -> ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) = <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
Theorem	axarch 8206*	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 9239 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 8246. (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 8207*	Peano's inductive postulate. This is a counterpart to peano5nni 9240 designed for real number axioms which involve natural numbers (notably, axcaucvg 8215). (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 8208*	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 9253 designed for real number axioms which involve natural numbers (notably, axcaucvg 8215). (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 8209*	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 8210*	Lemma for axcaucvg 8215. 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 8211*	Lemma for axcaucvg 8215. 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 8212*	Lemma for axcaucvg 8215. 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 8213*	Lemma for axcaucvg 8215. 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 8214*	Lemma for axcaucvg 8215. 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 8215*	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 8247. (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 8216*	Lemma for axpre-suploc 8217. 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 8124. (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 8217*	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 8248. (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 8218	The complex numbers form a set. Proofs should normally use cnex 8251 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
|- CC e. _V
Axiom	ax-resscn 8219	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 8175. (Contributed by NM, 1-Mar-1995.)
|- RR C_ CC
Axiom	ax-1cn 8220	1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 8176. (Contributed by NM, 1-Mar-1995.)
|- 1 e. CC
Axiom	ax-1re 8221	1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 8177. Proofs should use 1re 8273 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
|- 1 e. RR
Axiom	ax-icn 8222	_i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 8178. (Contributed by NM, 1-Mar-1995.)
|- _i e. CC
Axiom	ax-addcl 8223	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 8179. Proofs should normally use addcl 8252 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 8224	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 8180. Proofs should normally use readdcl 8253 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Axiom	ax-mulcl 8225	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 8181. Proofs should normally use mulcl 8254 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 8226	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 8182. Proofs should normally use remulcl 8255 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 8227	Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 8185. Proofs should normally use addcom 8410 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 8228	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8186. Proofs should normally use mulcom 8256 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 8229	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8187. Proofs should normally use addass 8257 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 8230	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8188. Proofs should normally use mulass 8258 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 8231	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 8189. Proofs should normally use adddi 8259 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 8232	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 8190. (Contributed by NM, 29-Jan-1995.)
|- ( ( _i x. _i ) + 1 ) = 0
Axiom	ax-0lt1 8233	0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 8191. Proofs should normally use 0lt1 8400 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
|- 0 <RR 1
Axiom	ax-1rid 8234	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 8192. (Contributed by NM, 29-Jan-1995.)
|- ( A e. RR -> ( A x. 1 ) = A )
Axiom	ax-0id 8235	0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 8193. Proofs should normally use addrid 8411 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Axiom	ax-rnegex 8236*	Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 8194. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Axiom	ax-precex 8237*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 8195. (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 8238*	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 8196. For naming consistency, use cnre 8270 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 8239	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 8239. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( A e. RR -> -. A <RR A )
Axiom	ax-pre-ltwlin 8240	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 8198. (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 8241	Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 8199. (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 8242	Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 8200. (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 8243	Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 8201. (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 8244	The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 8202. (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 8245	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, justified by Theorem axpre-mulext 8203 (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 8246*	Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by Theorem axarch 8206. This axiom should not be used directly; instead use arch 9493 (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 8247*	Completeness. Axiom for real and complex numbers, justified by Theorem axcaucvg 8215. 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 11666 (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 8248*	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 8247 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 8247. (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 8249	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 8252 should be used. Note that uses of ax-addf 8249 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 8183. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
|- + : ( CC X. CC ) --> CC
Axiom	ax-mulf 8250	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 8475 or eff 12349. 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 8254. Note that uses of ax-mulf 8250 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 8264. This axiom is justified by Theorem axmulf 8184. (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 8251	Alias for ax-cnex 8218. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- CC e. _V
Theorem	addcl 8252	Alias for ax-addcl 8223, for naming consistency with addcli 8278. Use this theorem instead of ax-addcl 8223 or axaddcl 8179. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	readdcl 8253	Alias for ax-addrcl 8224, for naming consistency with readdcli 8287. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	mulcl 8254	Alias for ax-mulcl 8225, for naming consistency with mulcli 8279. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	remulcl 8255	Alias for ax-mulrcl 8226, for naming consistency with remulcli 8288. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	mulcom 8256	Alias for ax-mulcom 8228, for naming consistency with mulcomi 8280. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	addass 8257	Alias for ax-addass 8229, for naming consistency with addassi 8282. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulass 8258	Alias for ax-mulass 8230, for naming consistency with mulassi 8283. (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 8259	Alias for ax-distr 8231, for naming consistency with adddii 8284. (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 8260	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
|- ( A e. RR -> A e. CC )
Theorem	reex 8261	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- RR e. _V
Theorem	reelprrecn 8262	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 8263	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 8264*	Multiplication is an operation on complex numbers. Version of ax-mulf 8250 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8225. (Contributed by GG, 16-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC
Theorem	adddir 8265	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 8266	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
|- 0 e. CC
Theorem	0cnd 8267	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. CC )
Theorem	c0ex 8268	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 e. _V
Theorem	1ex 8269	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 1 e. _V
Theorem	cnre 8270*	Alias for ax-cnre 8238, 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 8271	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 8272	Identity law for multiplication. Note: see mulrid 8271 for commuted version. (Contributed by NM, 8-Oct-1999.)
|- ( A e. CC -> ( 1 x. A ) = A )
Theorem	1re 8273	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
|- 1 e. RR
Theorem	0re 8274	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
|- 0 e. RR
Theorem	0red 8275	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 0 e. RR )
Theorem	mulridi 8276	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( A x. 1 ) = A
Theorem	mullidi 8277	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( 1 x. A ) = A
Theorem	addcli 8278	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) e. CC
Theorem	mulcli 8279	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) e. CC
Theorem	mulcomi 8280	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) = ( B x. A )
Theorem	mulcomli 8281	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 8282	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 8283	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 8284	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 8285	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 8286	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
|- A e. RR   =>    |- A e. CC
Theorem	readdcli 8287	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A + B ) e. RR
Theorem	remulcli 8288	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 8289	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 8290	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulridd 8291	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mullidd 8292	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 8293	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 8294	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 8295	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 8296	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 8297	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 8298	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 8299	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 8300	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 ) )