P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	caucvgsrlemfv 8001*	Lemma for caucvgsr 8012. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   &    |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )   =>    |- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )
Theorem	caucvgsrlemf 8002*	Lemma for caucvgsr 8012. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   &    |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )   =>    |- ( ph -> G : N. --> P. )
Theorem	caucvgsrlemcau 8003*	Lemma for caucvgsr 8012. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   &    |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )   =>    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
Theorem	caucvgsrlembound 8004*	Lemma for caucvgsr 8012. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   &    |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )   =>    |- ( ph -> A. m e. N. 1P <P ( G ` m ) )
Theorem	caucvgsrlemgt1 8005*	Lemma for caucvgsr 8012. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   =>    |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) )
Theorem	caucvgsrlemoffval 8006*	Lemma for caucvgsr 8012. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   &    |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )   =>    |- ( ( ph /\ J e. N. ) -> ( ( G ` J ) +R A ) = ( ( F ` J ) +R 1R ) )
Theorem	caucvgsrlemofff 8007*	Lemma for caucvgsr 8012. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   &    |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )   =>    |- ( ph -> G : N. --> R. )
Theorem	caucvgsrlemoffcau 8008*	Lemma for caucvgsr 8012. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   &    |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )   =>    |- ( 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	caucvgsrlemoffgt1 8009*	Lemma for caucvgsr 8012. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   &    |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )   =>    |- ( ph -> A. m e. N. 1R <R ( G ` m ) )
Theorem	caucvgsrlemoffres 8010*	Lemma for caucvgsr 8012. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   &    |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )   =>    |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( y +R x ) /\ y <R ( ( F ` k ) +R x ) ) ) ) )
Theorem	caucvgsrlembnd 8011*	Lemma for caucvgsr 8012. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   &    |- ( ph -> A. m e. N. A <R ( F ` m ) )   =>    |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( y +R x ) /\ y <R ( ( F ` k ) +R x ) ) ) ) )
Theorem	caucvgsr 8012*	A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. 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). This is similar to caucvgprpr 7922 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 8011). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 8007). 3. Since a signed real (element of R.) which is greater than zero can be mapped to a positive real (element of P.), perform that mapping on each element of the sequence and invoke caucvgprpr 7922 to get a limit (see caucvgsrlemgt1 8005). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8005). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8010). (Contributed by Jim Kingdon, 20-Jun-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )   =>    |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( y +R x ) /\ y <R ( ( F ` k ) +R x ) ) ) ) )
Theorem	ltpsrprg 8013	Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
|- ( ( A e. P. /\ B e. P. /\ C e. R. ) -> ( ( C +R [ <. A , 1P >. ] ~R ) <R ( C +R [ <. B , 1P >. ] ~R ) <-> A <P B ) )
Theorem	mappsrprg 8014	Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
|- ( ( A e. P. /\ C e. R. ) -> ( C +R -1R ) <R ( C +R [ <. A , 1P >. ] ~R ) )
Theorem	map2psrprg 8015*	Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
|- ( C e. R. -> ( ( C +R -1R ) <R A <-> E. x e. P. ( C +R [ <. x , 1P >. ] ~R ) = A ) )
Theorem	suplocsrlemb 8016*	Lemma for suplocsr 8019. The set B is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> A. u e. P. A. v e. P. ( u <P v -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )
Theorem	suplocsrlempr 8017*	Lemma for suplocsr 8019. The set B has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. v e. P. ( A. w e. B -. v <P w /\ A. w e. P. ( w <P v -> E. u e. B w <P u ) ) )
Theorem	suplocsrlem 8018*	Lemma for suplocsr 8019. The set A has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. x e. R. ( A. y e. A -. x <R y /\ A. y e. R. ( y <R x -> E. z e. A y <R z ) ) )
Theorem	suplocsr 8019*	An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. x e. R. ( A. y e. A -. x <R y /\ A. y e. R. ( y <R x -> E. z e. A y <R z ) ) )
Syntax	cc 8020	Class of complex numbers.
class CC
Syntax	cr 8021	Class of real numbers.
class RR
Syntax	cc0 8022	Extend class notation to include the complex number 0.
class 0
Syntax	c1 8023	Extend class notation to include the complex number 1.
class 1
Syntax	ci 8024	Extend class notation to include the complex number i.
class _i
Syntax	caddc 8025	Addition on complex numbers.
class +
Syntax	cltrr 8026	'Less than' predicate (defined over real subset of complex numbers).
class <RR
Syntax	cmul 8027	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 8028	Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- CC = ( R. X. R. )
Definition	df-0 8029	Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
|- 0 = <. 0R , 0R >.
Definition	df-1 8030	Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
|- 1 = <. 1R , 0R >.
Definition	df-i 8031	Define the complex number _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
|- _i = <. 0R , 1R >.
Definition	df-r 8032	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
|- RR = ( R. X. { 0R } )
Definition	df-add 8033*	Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
|- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
Definition	df-mul 8034*	Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
|- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
Definition	df-lt 8035*	Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) }
Theorem	opelcn 8036	Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
|- ( <. A , B >. e. CC <-> ( A e. R. /\ B e. R. ) )
Theorem	opelreal 8037	Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- ( <. A , 0R >. e. RR <-> A e. R. )
Theorem	elreal 8038*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
|- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
Theorem	elrealeu 8039*	The real number mapping in elreal 8038 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
Theorem	elreal2 8040	Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
|- ( A e. RR <-> ( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) )
Theorem	0ncn 8041	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 8042 which is a related property. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	cnm 8042*	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 8043	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 8044	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 8045	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 8046	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 8047	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 8048	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 8049	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 8050	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 8051	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6764, 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 8028), 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 8052	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8051 and mulcnsrec 8053. (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 8053	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6763, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8051. (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 8054	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 8147. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
Theorem	pitonnlem1 8055*	Lemma for pitonn 8058. 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 8056	Lemma for pitonn 8058. 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 8057*	Lemma for pitonn 8058. 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 8058*	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 8059*	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 8060*	Embedding from N. to RR. Similar to pitonn 8058 but separate in the sense that we have not proved nnssre 9137 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 8061*	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 8062*	One is an element of NN. This is a counterpart to 1nn 9144 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. N
Theorem	peano2nnnn 8063*	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 9145 designed for real number axioms which involve to 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 ) }   =>    |- ( A e. N -> ( A + 1 ) e. N )
Theorem	ltrennb 8064*	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 8065*	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 8066*	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 8067	Lemma for recidpirq 8068. 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 8068*	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 8069	The complex numbers form a set. Use cnex 8146 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 8070	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 8114. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 8071	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 8115. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	ax1re 8072	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 8116. In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 8115 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 8073	_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 8117. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 8074	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 8118 be used later. Instead, in most cases use addcl 8147. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 8075	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 8119 be used later. Instead, in most cases use readdcl 8148. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 8076	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 8120 be used later. Instead, in most cases use mulcl 8149. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 8077	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 8121 be used later. Instead, in most cases use remulcl 8150. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	axaddf 8078	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 8074. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8144. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- + : ( CC X. CC ) --> CC
Theorem	axmulf 8079	Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 8145 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 8149. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- x. : ( CC X. CC ) --> CC
Theorem	axaddcom 8080	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 8122 be used later. Instead, use addcom 8306. 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 8081	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 8123 be used later. Instead, use mulcom 8151. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	axaddass 8082	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 8124 be used later. Instead, use addass 8152. (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 8083	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 8125. (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 8084	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 8126 be used later. Instead, use adddi 8154. (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 8085	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 8127. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( ( _i x. _i ) + 1 ) = 0
Theorem	ax0lt1 8086	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 8128. 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 8087	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 8129. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
|- ( A e. RR -> ( A x. 1 ) = A )
Theorem	ax0id 8088	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 8130. 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 8089*	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 8131. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Theorem	axprecex 8090*	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 8132. 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 8091*	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 8133. (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 8092	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 8134. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- ( A e. RR -> -. A <RR A )
Theorem	axpre-ltwlin 8093	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 8135. (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 8094	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 8136. (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 8095	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 8137. (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 8096	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 8138. (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 8097	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 8139. (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 8098	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 8140. (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 8099*	The reciprocal from axprecex 8090 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 8100*	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 >. )