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	1idsr 7901	1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
|- ( A e. R. -> ( A .R 1R ) = A )
Theorem	00sr 7902	A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
|- ( A e. R. -> ( A .R 0R ) = 0R )
Theorem	ltasrg 7903	Ordering property of addition. (Contributed by NM, 10-May-1996.)
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A <R B <-> ( C +R A ) <R ( C +R B ) ) )
Theorem	pn0sr 7904	A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
|- ( A e. R. -> ( A +R ( A .R -1R ) ) = 0R )
Theorem	negexsr 7905*	Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
|- ( A e. R. -> E. x e. R. ( A +R x ) = 0R )
Theorem	recexgt0sr 7906*	The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( 0R <R A -> E. x e. R. ( 0R <R x /\ ( A .R x ) = 1R ) )
Theorem	recexsrlem 7907*	The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
|- ( 0R <R A -> E. x e. R. ( A .R x ) = 1R )
Theorem	addgt0sr 7908	The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
|- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A +R B ) )
Theorem	ltadd1sr 7909	Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
|- ( A e. R. -> A <R ( A +R 1R ) )
Theorem	ltm1sr 7910	Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
|- ( A e. R. -> ( A +R -1R ) <R A )
Theorem	mulgt0sr 7911	The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
|- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) )
Theorem	aptisr 7912	Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )
Theorem	mulextsr1lem 7913	Lemma for mulextsr1 7914. (Contributed by Jim Kingdon, 17-Feb-2020.)
|- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) -> ( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) )
Theorem	mulextsr1 7914	Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) )
Theorem	archsr 7915*	For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q }, { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R is the embedding of the positive integer x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
|- ( A e. R. -> E. x e. N. A <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
Theorem	srpospr 7916*	Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
|- ( ( A e. R. /\ 0R <R A ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )
Theorem	prsrcl 7917	Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
|- ( A e. P. -> [ <. ( A +P. 1P ) , 1P >. ] ~R e. R. )
Theorem	prsrpos 7918	Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
|- ( A e. P. -> 0R <R [ <. ( A +P. 1P ) , 1P >. ] ~R )
Theorem	prsradd 7919	Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
|- ( ( A e. P. /\ B e. P. ) -> [ <. ( ( A +P. B ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( A +P. 1P ) , 1P >. ] ~R +R [ <. ( B +P. 1P ) , 1P >. ] ~R ) )
Theorem	prsrlt 7920	Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
|- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R ) )
Theorem	prsrriota 7921*	Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
|- ( ( A e. R. /\ 0R <R A ) -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A )
Theorem	caucvgsrlemcl 7922*	Lemma for caucvgsr 7935. Terms of the sequence from caucvgsrlemgt1 7928 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
|- ( ph -> F : N. --> R. )   &    |- ( ph -> A. m e. N. 1R <R ( F ` m ) )   =>    |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. )
Theorem	caucvgsrlemasr 7923*	Lemma for caucvgsr 7935. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
|- ( ph -> A. m e. N. A <R ( F ` m ) )   =>    |- ( ph -> A e. R. )
Theorem	caucvgsrlemfv 7924*	Lemma for caucvgsr 7935. 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 7925*	Lemma for caucvgsr 7935. 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 7926*	Lemma for caucvgsr 7935. 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 7927*	Lemma for caucvgsr 7935. 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 7928*	Lemma for caucvgsr 7935. 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 7929*	Lemma for caucvgsr 7935. 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 7930*	Lemma for caucvgsr 7935. 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 7931*	Lemma for caucvgsr 7935. 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 7932*	Lemma for caucvgsr 7935. 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 7933*	Lemma for caucvgsr 7935. 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 7934*	Lemma for caucvgsr 7935. 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 7935*	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 7845 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 7934). 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 7930). 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 7845 to get a limit (see caucvgsrlemgt1 7928). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7928). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7933). (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 7936	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 7937	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 7938*	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 7939*	Lemma for suplocsr 7942. 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 7940*	Lemma for suplocsr 7942. 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 7941*	Lemma for suplocsr 7942. 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 7942*	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 7943	Class of complex numbers.
class CC
Syntax	cr 7944	Class of real numbers.
class RR
Syntax	cc0 7945	Extend class notation to include the complex number 0.
class 0
Syntax	c1 7946	Extend class notation to include the complex number 1.
class 1
Syntax	ci 7947	Extend class notation to include the complex number i.
class _i
Syntax	caddc 7948	Addition on complex numbers.
class +
Syntax	cltrr 7949	'Less than' predicate (defined over real subset of complex numbers).
class <RR
Syntax	cmul 7950	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 7951	Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- CC = ( R. X. R. )
Definition	df-0 7952	Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
|- 0 = <. 0R , 0R >.
Definition	df-1 7953	Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
|- 1 = <. 1R , 0R >.
Definition	df-i 7954	Define the complex number _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
|- _i = <. 0R , 1R >.
Definition	df-r 7955	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
|- RR = ( R. X. { 0R } )
Definition	df-add 7956*	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 7957*	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 7958*	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 7959	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 7960	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 7961*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
|- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
Theorem	elrealeu 7962*	The real number mapping in elreal 7961 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
Theorem	elreal2 7963	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 7964	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 7965 which is a related property. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	cnm 7965*	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 7966	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 7967	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 7968	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 7969	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 7970	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 7971	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 7972	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 7973	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 7974	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6700, 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 7951), 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 7975	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7974 and mulcnsrec 7976. (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 7976	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6699, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7974. (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 7977	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 8070. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
Theorem	pitonnlem1 7978*	Lemma for pitonn 7981. 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 7979	Lemma for pitonn 7981. 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 7980*	Lemma for pitonn 7981. 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 7981*	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 7982*	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 7983*	Embedding from N. to RR. Similar to pitonn 7981 but separate in the sense that we have not proved nnssre 9060 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 7984*	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 7985*	One is an element of NN. This is a counterpart to 1nn 9067 designed for real number axioms which involve natural numbers (notably, axcaucvg 8033). (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 7986*	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 9068 designed for real number axioms which involve to natural numbers (notably, axcaucvg 8033). (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 7987*	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 7988*	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 7989*	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 7990	Lemma for recidpirq 7991. 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 7991*	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 7992	The complex numbers form a set. Use cnex 8069 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 7993	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 8037. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 7994	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 8038. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	ax1re 7995	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 8039. In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 8038 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 7996	_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 8040. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 7997	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 8041 be used later. Instead, in most cases use addcl 8070. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 7998	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 8042 be used later. Instead, in most cases use readdcl 8071. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 7999	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 8043 be used later. Instead, in most cases use mulcl 8072. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 8000	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 8044 be used later. Instead, in most cases use remulcl 8073. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )