P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0ncn 7901	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. See also cnm 7902 which is a related property. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	cnm 7902*	A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
|- ( A e. CC -> E. x x e. A )
Theorem	ltrelre 7903	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 7904	Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.)
|- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. )
Theorem	mulcnsr 7905	Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.)
|- ( ( ( 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 7906	Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
|- A e. _V   =>    |- ( <. A , 0R >. = <. B , 0R >. <-> A = B )
Theorem	addresr 7907	Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
|- ( ( A e. R. /\ B e. R. ) -> ( <. A , 0R >. + <. B , 0R >. ) = <. ( A +R B ) , 0R >. )
Theorem	mulresr 7908	Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
|- ( ( A e. R. /\ B e. R. ) -> ( <. A , 0R >. x. <. B , 0R >. ) = <. ( A .R B ) , 0R >. )
Theorem	ltresr 7909	Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
|- ( <. A , 0R >. <RR <. B , 0R >. <-> A <R B )
Theorem	ltresr2 7910	Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> ( 1st ` A ) <R ( 1st ` B ) ) )
Theorem	dfcnqs 7911	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6661, 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 7888), 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.)
|- CC = ( ( R. X. R. ) /. `' _E )
Theorem	addcnsrec 7912	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7911 and mulcnsrec 7913. (Contributed by NM, 13-Aug-1995.)
|- ( ( ( 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 7913	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6660, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7911. (Contributed by NM, 13-Aug-1995.)
|- ( ( ( 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 )
Theorem	addvalex 7914	Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 8007. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
Theorem	pitonnlem1 7915*	Lemma for pitonn 7918. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
|- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1
Theorem	pitonnlem1p1 7916	Lemma for pitonn 7918. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
|- ( A e. P. -> [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( A +P. 1P ) , 1P >. ] ~R )
Theorem	pitonnlem2 7917*	Lemma for pitonn 7918. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
|- ( K e. N. -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
Theorem	pitonn 7918*	Mapping from N. to NN. (Contributed by Jim Kingdon, 22-Apr-2020.)
|- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
Theorem	pitoregt0 7919*	Embedding from N. to RR yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> 0 <RR <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
Theorem	pitore 7920*	Embedding from N. to RR. Similar to pitonn 7918 but separate in the sense that we have not proved nnssre 8997 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. RR )
Theorem	recnnre 7921*	Embedding the reciprocal of a natural number into RR. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. RR )
Theorem	peano1nnnn 7922*	One is an element of NN. This is a counterpart to 1nn 9004 designed for real number axioms which involve natural numbers (notably, axcaucvg 7970). (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. N
Theorem	peano2nnnn 7923*	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 9005 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7970). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( A e. N -> ( A + 1 ) e. N )
Theorem	ltrennb 7924*	Ordering of natural numbers with <N or <RR. (Contributed by Jim Kingdon, 13-Jul-2021.)
|- ( ( J e. N. /\ K e. N. ) -> ( J <N K <-> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
Theorem	ltrenn 7925*	Ordering of natural numbers with <N or <RR. (Contributed by Jim Kingdon, 12-Jul-2021.)
|- ( J <N K -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
Theorem	recidpipr 7926*	Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
|- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P )
Theorem	recidpirqlemcalc 7927	Lemma for recidpirq 7928. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
|- ( ph -> A e. P. )   &    |- ( ph -> B e. P. )   &    |- ( ph -> ( A .P. B ) = 1P )   =>    |- ( ph -> ( ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) +P. 1P ) = ( ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) )
Theorem	recidpirq 7928*	A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = 1 )
4.1.2  Final derivation of real and complex number postulates
Theorem	axcnex 7929	The complex numbers form a set. Use cnex 8006 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 7930	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 7974. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 7931	1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 7975. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	ax1re 7932	1 is a real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1re 7976. In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7975 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
|- 1 e. RR
Theorem	axicn 7933	_i is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 7977. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 7934	Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 7978 be used later. Instead, in most cases use addcl 8007. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 7935	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 7979 be used later. Instead, in most cases use readdcl 8008. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 7936	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 7980 be used later. Instead, in most cases use mulcl 8009. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 7937	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 7981 be used later. Instead, in most cases use remulcl 8010. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	axaddf 7938	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 7934. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8004. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- + : ( CC X. CC ) --> CC
Theorem	axmulf 7939	Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 8005 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 8009. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- x. : ( CC X. CC ) --> CC
Theorem	axaddcom 7940	Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 7982 be used later. Instead, use addcom 8166. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	axmulcom 7941	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7983 be used later. Instead, use mulcom 8011. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	axaddass 7942	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 for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 7984 be used later. Instead, use addass 8012. (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 7943	Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 7985. (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 7944	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 7986 be used later. Instead, use adddi 8014. (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 7945	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 7987. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( ( _i x. _i ) + 1 ) = 0
Theorem	ax0lt1 7946	0 is less than 1. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0lt1 7988. The version of this axiom in the Metamath Proof Explorer reads 1 =/= 0; here we change it to 0 <RR 1. The proof of 0 <RR 1 from 1 =/= 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- 0 <RR 1
Theorem	ax1rid 7947	1 is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7989. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
|- ( A e. RR -> ( A x. 1 ) = A )
Theorem	ax0id 7948	0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 7990. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	axrnegex 7949*	Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7991. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Theorem	axprecex 7950*	Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 7992. In treatments which assume excluded middle, the 0 <RR A condition is generally replaced by A =/= 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
Theorem	axcnre 7951*	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, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7993. (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-ltirr 7952	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7994. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- ( A e. RR -> -. A <RR A )
Theorem	axpre-ltwlin 7953	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7995. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
Theorem	axpre-lttrn 7954	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7996. (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-apti 7955	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 7997. (Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
Theorem	axpre-ltadd 7956	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 7998. (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 7957	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 7999. (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 7958	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 8000. (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 7959*	The reciprocal from axprecex 7950 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 7960*	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 7961*	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 8995 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 8001. (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 7962*	Peano's inductive postulate. This is a counterpart to peano5nni 8996 designed for real number axioms which involve natural numbers (notably, axcaucvg 7970). (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 7963*	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 9009 designed for real number axioms which involve natural numbers (notably, axcaucvg 7970). (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 7964*	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 7965*	Lemma for axcaucvg 7970. 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 7966*	Lemma for axcaucvg 7970. 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 7967*	Lemma for axcaucvg 7970. 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 7968*	Lemma for axcaucvg 7970. 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 7969*	Lemma for axcaucvg 7970. 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 7970*	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 8002. (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 7971*	Lemma for axpre-suploc 7972. 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 7879. (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 7972*	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 8003. (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 7973	The complex numbers form a set. Proofs should normally use cnex 8006 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
|- CC e. _V
Axiom	ax-resscn 7974	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 7930. (Contributed by NM, 1-Mar-1995.)
|- RR C_ CC
Axiom	ax-1cn 7975	1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 7931. (Contributed by NM, 1-Mar-1995.)
|- 1 e. CC
Axiom	ax-1re 7976	1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 7932. Proofs should use 1re 8028 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
|- 1 e. RR
Axiom	ax-icn 7977	_i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 7933. (Contributed by NM, 1-Mar-1995.)
|- _i e. CC
Axiom	ax-addcl 7978	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 7934. Proofs should normally use addcl 8007 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 7979	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 7935. Proofs should normally use readdcl 8008 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Axiom	ax-mulcl 7980	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7936. Proofs should normally use mulcl 8009 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 7981	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7937. Proofs should normally use remulcl 8010 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 7982	Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 7940. Proofs should normally use addcom 8166 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 7983	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 7941. Proofs should normally use mulcom 8011 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 7984	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7942. Proofs should normally use addass 8012 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 7985	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7943. Proofs should normally use mulass 8013 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 7986	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 7944. Proofs should normally use adddi 8014 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 7987	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 7945. (Contributed by NM, 29-Jan-1995.)
|- ( ( _i x. _i ) + 1 ) = 0
Axiom	ax-0lt1 7988	0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 7946. Proofs should normally use 0lt1 8156 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
|- 0 <RR 1
Axiom	ax-1rid 7989	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 7947. (Contributed by NM, 29-Jan-1995.)
|- ( A e. RR -> ( A x. 1 ) = A )
Axiom	ax-0id 7990	0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 7948. Proofs should normally use addrid 8167 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Axiom	ax-rnegex 7991*	Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 7949. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Axiom	ax-precex 7992*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 7950. (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 7993*	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 7951. For naming consistency, use cnre 8025 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 7994	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 7994. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( A e. RR -> -. A <RR A )
Axiom	ax-pre-ltwlin 7995	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7953. (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 7996	Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 7954. (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 7997	Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7955. (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 7998	Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 7956. (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 7999	The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 7957. (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 8000	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, justified by Theorem axpre-mulext 7958 (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 ) ) )