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	ltrelsr 8001	Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
|- <R C_ ( R. X. R. )
Theorem	addcmpblnr 8002	Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( A +P. F ) , ( B +P. G ) >. ~R <. ( C +P. R ) , ( D +P. S ) >. ) )
Theorem	mulcmpblnrlemg 8003	Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> ( ( D .P. F ) +P. ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) ) = ( ( D .P. F ) +P. ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) ) )
Theorem	mulcmpblnr 8004	Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) )
Theorem	prsrlem1 8005*	Decomposing signed reals into positive reals. Lemma for addsrpr 8008 and mulsrpr 8009. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) )
Theorem	addsrmo 8006*	There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
Theorem	mulsrmo 8007*	There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
Theorem	addsrpr 8008	Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
Theorem	mulsrpr 8009	Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
Theorem	ltsrprg 8010	Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )
Theorem	gt0srpr 8011	Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
|- ( 0R <R [ <. A , B >. ] ~R <-> B <P A )
Theorem	0nsr 8012	The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
|- -. (/) e. R.
Theorem	0r 8013	The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.)
|- 0R e. R.
Theorem	1sr 8014	The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.)
|- 1R e. R.
Theorem	m1r 8015	The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
|- -1R e. R.
Theorem	addclsr 8016	Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
|- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) e. R. )
Theorem	mulclsr 8017	Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
|- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) e. R. )
Theorem	addcomsrg 8018	Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) = ( B +R A ) )
Theorem	addasssrg 8019	Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A +R B ) +R C ) = ( A +R ( B +R C ) ) )
Theorem	mulcomsrg 8020	Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )
Theorem	mulasssrg 8021	Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) )
Theorem	distrsrg 8022	Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) )
Theorem	m1p1sr 8023	Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
|- ( -1R +R 1R ) = 0R
Theorem	m1m1sr 8024	Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
|- ( -1R .R -1R ) = 1R
Theorem	lttrsr 8025*	Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
|- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f <R g /\ g <R h ) -> f <R h ) )
Theorem	ltposr 8026	Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
|- <R Po R.
Theorem	ltsosr 8027	Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
|- <R Or R.
Theorem	0lt1sr 8028	0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
|- 0R <R 1R
Theorem	1ne0sr 8029	1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
|- -. 1R = 0R
Theorem	0idsr 8030	The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
|- ( A e. R. -> ( A +R 0R ) = A )
Theorem	1idsr 8031	1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
|- ( A e. R. -> ( A .R 1R ) = A )
Theorem	00sr 8032	A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
|- ( A e. R. -> ( A .R 0R ) = 0R )
Theorem	ltasrg 8033	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 8034	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 8035*	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 8036*	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 8037*	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 8038	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 8039	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 8040	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 8041	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 8042	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 8043	Lemma for mulextsr1 8044. (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 8044	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 8045*	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 8046*	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 8047	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 8048	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 8049	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 8050	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 8051*	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 8052*	Lemma for caucvgsr 8065. Terms of the sequence from caucvgsrlemgt1 8058 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 8053*	Lemma for caucvgsr 8065. 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 8054*	Lemma for caucvgsr 8065. 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 8055*	Lemma for caucvgsr 8065. 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 8056*	Lemma for caucvgsr 8065. 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 8057*	Lemma for caucvgsr 8065. 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 8058*	Lemma for caucvgsr 8065. 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 8059*	Lemma for caucvgsr 8065. 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 8060*	Lemma for caucvgsr 8065. 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 8061*	Lemma for caucvgsr 8065. 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 8062*	Lemma for caucvgsr 8065. 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 8063*	Lemma for caucvgsr 8065. 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 8064*	Lemma for caucvgsr 8065. 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 8065*	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 7975 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 8064). 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 8060). 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 7975 to get a limit (see caucvgsrlemgt1 8058). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8058). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8063). (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 8066	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 8067	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 8068*	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 8069*	Lemma for suplocsr 8072. 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 8070*	Lemma for suplocsr 8072. 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 8071*	Lemma for suplocsr 8072. 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 8072*	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 8073	Class of complex numbers.
class CC
Syntax	cr 8074	Class of real numbers.
class RR
Syntax	cc0 8075	Extend class notation to include the complex number 0.
class 0
Syntax	c1 8076	Extend class notation to include the complex number 1.
class 1
Syntax	ci 8077	Extend class notation to include the complex number i.
class _i
Syntax	caddc 8078	Addition on complex numbers.
class +
Syntax	cltrr 8079	'Less than' predicate (defined over real subset of complex numbers).
class <RR
Syntax	cmul 8080	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 8081	Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
|- CC = ( R. X. R. )
Definition	df-0 8082	Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
|- 0 = <. 0R , 0R >.
Definition	df-1 8083	Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
|- 1 = <. 1R , 0R >.
Definition	df-i 8084	Define the complex number _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
|- _i = <. 0R , 1R >.
Definition	df-r 8085	Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
|- RR = ( R. X. { 0R } )
Definition	df-add 8086*	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 8087*	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 8088*	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 8089	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 8090	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 8091*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
|- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
Theorem	elrealeu 8092*	The real number mapping in elreal 8091 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
Theorem	elreal2 8093	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 8094	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 8095 which is a related property. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	cnm 8095*	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 8096	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 8097	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 8098	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 8099	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 8100	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 >. )