P. 82 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	ltasrg 8101	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 8102	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 8103*	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 8104*	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 8105*	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 8106	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 8107	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 8108	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 8109	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 8110	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 8111	Lemma for mulextsr1 8112. (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 8112	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 8113*	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 8114*	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 8115	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 8116	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 8117	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 8118	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 8119*	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 8120*	Lemma for caucvgsr 8133. Terms of the sequence from caucvgsrlemgt1 8126 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 8121*	Lemma for caucvgsr 8133. 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 8122*	Lemma for caucvgsr 8133. 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 8123*	Lemma for caucvgsr 8133. 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 8124*	Lemma for caucvgsr 8133. 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 8125*	Lemma for caucvgsr 8133. 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 8126*	Lemma for caucvgsr 8133. 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 8127*	Lemma for caucvgsr 8133. 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 8128*	Lemma for caucvgsr 8133. 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 8129*	Lemma for caucvgsr 8133. 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 8130*	Lemma for caucvgsr 8133. 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 8131*	Lemma for caucvgsr 8133. 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 8132*	Lemma for caucvgsr 8133. 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 8133*	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 8043 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 8132). 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 8128). 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 8043 to get a limit (see caucvgsrlemgt1 8126). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8126). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8131). (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 8134	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 8135	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 8136*	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 8137*	Lemma for suplocsr 8140. 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 8138*	Lemma for suplocsr 8140. 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 8139*	Lemma for suplocsr 8140. 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 8140*	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 8141	Class of complex numbers.
class CC
Syntax	cr 8142	Class of real numbers.
class RR
Syntax	cc0 8143	Extend class notation to include the complex number 0.
class 0
Syntax	c1 8144	Extend class notation to include the complex number 1.
class 1
Syntax	ci 8145	Extend class notation to include the complex number i.
class _i
Syntax	caddc 8146	Addition on complex numbers.
class +
Syntax	cltrr 8147	'Less than' predicate (defined over real subset of complex numbers).
class <RR
Syntax	cmul 8148	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 8149	Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- CC = ( R. X. R. )
Definition	df-0 8150	Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
|- 0 = <. 0R , 0R >.
Definition	df-1 8151	Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
|- 1 = <. 1R , 0R >.
Definition	df-i 8152	Define the complex number _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
|- _i = <. 0R , 1R >.
Definition	df-r 8153	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
|- RR = ( R. X. { 0R } )
Definition	df-add 8154*	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 8155*	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 8156*	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 8157	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 8158	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 8159*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
|- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
Theorem	elrealeu 8160*	The real number mapping in elreal 8159 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
Theorem	elreal2 8161	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 8162	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 8163 which is a related property. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	cnm 8163*	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 8164	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 8165	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 8166	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 8167	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 8168	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 8169	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 8170	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 8171	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 8172	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6847, 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 8149), 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 8173	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8172 and mulcnsrec 8174. (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 8174	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6846, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8172. (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 8175	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 8268. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
Theorem	pitonnlem1 8176*	Lemma for pitonn 8179. 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 8177	Lemma for pitonn 8179. 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 8178*	Lemma for pitonn 8179. 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 8179*	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 8180*	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 8181*	Embedding from N. to RR. Similar to pitonn 8179 but separate in the sense that we have not proved nnssre 9258 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 8182*	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 8183*	One is an element of NN. This is a counterpart to 1nn 9265 designed for real number axioms which involve natural numbers (notably, axcaucvg 8231). (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 8184*	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 9266 designed for real number axioms which involve to natural numbers (notably, axcaucvg 8231). (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 8185*	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 8186*	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 8187*	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 8188	Lemma for recidpirq 8189. 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 8189*	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 8190	The complex numbers form a set. Use cnex 8267 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 8191	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 8235. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 8192	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 8236. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	ax1re 8193	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 8237. In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 8236 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 8194	_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 8238. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 8195	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 8239 be used later. Instead, in most cases use addcl 8268. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 8196	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 8240 be used later. Instead, in most cases use readdcl 8269. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 8197	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 8241 be used later. Instead, in most cases use mulcl 8270. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 8198	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 8242 be used later. Instead, in most cases use remulcl 8271. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	axaddf 8199	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 8195. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8265. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- + : ( CC X. CC ) --> CC
Theorem	axmulf 8200	Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 8266 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 8270. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
|- x. : ( CC X. CC ) --> CC