P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opelcn 7301	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 7302	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 7303*	Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
|- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
Theorem	elrealeu 7304*	The real number mapping in elreal 7303 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
Theorem	elreal2 7305	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 7306	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. (Contributed by NM, 2-May-1996.)
|- -. (/) e. CC
Theorem	ltrelre 7307	'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
|- <RR C_ ( RR X. RR )
Theorem	addcnsr 7308	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 7309	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 7310	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 7311	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 7312	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 7313	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 7314	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 7315	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6303, 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 7293), 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 7316	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7315 and mulcnsrec 7317. (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 7317	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6302, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7315. (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 7318	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 7404. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
Theorem	pitonnlem1 7319*	Lemma for pitonn 7322. 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 7320	Lemma for pitonn 7322. 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 7321*	Lemma for pitonn 7322. 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 7322*	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 7323*	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 7324*	Embedding from N. to RR. Similar to pitonn 7322 but separate in the sense that we have not proved nnssre 8354 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 7325*	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 7326*	One is an element of NN. This is a counterpart to 1nn 8361 designed for real number axioms which involve natural numbers (notably, axcaucvg 7372). (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 7327*	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8362 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7372). (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 7328*	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 7329*	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 7330*	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 7331	Lemma for recidpirq 7332. 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 7332*	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 )
3.1.2  Final derivation of real and complex number postulates
Theorem	axcnex 7333	The complex numbers form a set. Use cnex 7403 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- CC e. _V
Theorem	axresscn 7334	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 7374. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
|- RR C_ CC
Theorem	ax1cn 7335	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 7375. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
|- 1 e. CC
Theorem	ax1re 7336	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 7376. In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7375 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 7337	_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 7377. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
|- _i e. CC
Theorem	axaddcl 7338	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 7378 be used later. Instead, in most cases use addcl 7404. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Theorem	axaddrcl 7339	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 7379 be used later. Instead, in most cases use readdcl 7405. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Theorem	axmulcl 7340	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 7380 be used later. Instead, in most cases use mulcl 7406. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	axmulrcl 7341	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 7381 be used later. Instead, in most cases use remulcl 7407. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	axaddcom 7342	Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 7382 be used later. Instead, use addcom 7556. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	axmulcom 7343	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7383 be used later. Instead, use mulcom 7408. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	axaddass 7344	Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 7384 be used later. Instead, use addass 7409. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	axmulass 7345	Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 7385. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
Theorem	axdistr 7346	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 7386 be used later. Instead, use adddi 7411. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
Theorem	axi2m1 7347	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 7387. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
|- ( ( _i x. _i ) + 1 ) = 0
Theorem	ax0lt1 7348	0 is less than 1. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0lt1 7388. The version of this axiom in the Metamath Proof Explorer reads 1 =/= 0; here we change it to 0 <RR 1. The proof of 0 <RR 1 from 1 =/= 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- 0 <RR 1
Theorem	ax1rid 7349	1 is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7389. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
|- ( A e. RR -> ( A x. 1 ) = A )
Theorem	ax0id 7350	0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 7390. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	axrnegex 7351*	Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7391. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Theorem	axprecex 7352*	Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 7392. In treatments which assume excluded middle, the 0 <RR A condition is generally replaced by A =/= 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
Theorem	axcnre 7353*	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7393. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
Theorem	axpre-ltirr 7354	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7394. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- ( A e. RR -> -. A <RR A )
Theorem	axpre-ltwlin 7355	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7395. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
Theorem	axpre-lttrn 7356	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7396. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
Theorem	axpre-apti 7357	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 7397. (Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
Theorem	axpre-ltadd 7358	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7398. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
Theorem	axpre-mulgt0 7359	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 7399. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
Theorem	axpre-mulext 7360	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7400. (Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) )
Theorem	rereceu 7361*	The reciprocal from axprecex 7352 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( ( A e. RR /\ 0 <RR A ) -> E! x e. RR ( A x. x ) = 1 )
Theorem	recriota 7362*	Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( N e. N. -> ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) = <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
Theorem	axarch 7363*	Archimedean axiom. The Archimedean property is more naturally stated once we have defined NN. Unless we find another way to state it, we'll just use the right hand side of dfnn2 8352 in stating what we mean by "natural number" in the context of this axiom. This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7401. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)
|- ( A e. RR -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )
Theorem	peano5nnnn 7364*	Peano's inductive postulate. This is a counterpart to peano5nni 8353 designed for real number axioms which involve natural numbers (notably, axcaucvg 7372). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A )
Theorem	nnindnn 7365*	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8366 designed for real number axioms which involve natural numbers (notably, axcaucvg 7372). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( z = 1 -> ( ph <-> ps ) )   &    |- ( z = k -> ( ph <-> ch ) )   &    |- ( z = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( z = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( k e. N -> ( ch -> th ) )   =>    |- ( A e. N -> ta )
Theorem	nntopi 7366*	Mapping from NN to N.. (Contributed by Jim Kingdon, 13-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( A e. N -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A )
Theorem	axcaucvglemcl 7367*	Lemma for axcaucvg 7372. Mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   =>    |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) e. R. )
Theorem	axcaucvglemf 7368*	Lemma for axcaucvg 7372. Mapping to N. and R. yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> G : N. --> R. )
Theorem	axcaucvglemval 7369*	Lemma for axcaucvg 7372. Value of sequence when mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. )
Theorem	axcaucvglemcau 7370*	Lemma for axcaucvg 7372. The result of mapping to N. and R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
Theorem	axcaucvglemres 7371*	Lemma for axcaucvg 7372. Mapping the limit from N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
Theorem	axcaucvg 7372*	Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). Because we are stating this axiom before we have introduced notations for NN or division, we use N for the natural numbers and express a reciprocal in terms of iota_. This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7402. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
3.1.3  Real and complex number postulates restated as axioms
Axiom	ax-cnex 7373	The complex numbers form a set. Proofs should normally use cnex 7403 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
|- CC e. _V
Axiom	ax-resscn 7374	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 7334. (Contributed by NM, 1-Mar-1995.)
|- RR C_ CC
Axiom	ax-1cn 7375	1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 7335. (Contributed by NM, 1-Mar-1995.)
|- 1 e. CC
Axiom	ax-1re 7376	1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 7336. Proofs should use 1re 7424 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
|- 1 e. RR
Axiom	ax-icn 7377	_i is a complex number. Axiom for real and complex numbers, justified by theorem axicn 7337. (Contributed by NM, 1-Mar-1995.)
|- _i e. CC
Axiom	ax-addcl 7378	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7338. Proofs should normally use addcl 7404 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
Axiom	ax-addrcl 7379	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 7339. Proofs should normally use readdcl 7405 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
Axiom	ax-mulcl 7380	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7340. Proofs should normally use mulcl 7406 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Axiom	ax-mulrcl 7381	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7341. Proofs should normally use remulcl 7407 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Axiom	ax-addcom 7382	Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7342. Proofs should normally use addcom 7556 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Axiom	ax-mulcom 7383	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7343. Proofs should normally use mulcom 7408 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Axiom	ax-addass 7384	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7344. Proofs should normally use addass 7409 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Axiom	ax-mulass 7385	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7345. Proofs should normally use mulass 7410 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
Axiom	ax-distr 7386	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7346. Proofs should normally use adddi 7411 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
Axiom	ax-i2m1 7387	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7347. (Contributed by NM, 29-Jan-1995.)
|- ( ( _i x. _i ) + 1 ) = 0
Axiom	ax-0lt1 7388	0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7348. Proofs should normally use 0lt1 7547 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
|- 0 <RR 1
Axiom	ax-1rid 7389	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7349. (Contributed by NM, 29-Jan-1995.)
|- ( A e. RR -> ( A x. 1 ) = A )
Axiom	ax-0id 7390	0 is an identity element for real addition. Axiom for real and complex numbers, justified by theorem ax0id 7350. Proofs should normally use addid1 7557 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Axiom	ax-rnegex 7391*	Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7351. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
Axiom	ax-precex 7392*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7352. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
Axiom	ax-cnre 7393*	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by theorem axcnre 7353. For naming consistency, use cnre 7421 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
Axiom	ax-pre-ltirr 7394	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7394. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( A e. RR -> -. A <RR A )
Axiom	ax-pre-ltwlin 7395	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7355. (Contributed by Jim Kingdon, 12-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )
Axiom	ax-pre-lttrn 7396	Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7356. (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )
Axiom	ax-pre-apti 7397	Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7357. (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )
Axiom	ax-pre-ltadd 7398	Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 7358. (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
Axiom	ax-pre-mulgt0 7399	The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7359. (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
Axiom	ax-pre-mulext 7400	Strong extensionality of multiplication (expressed in terms of <RR). Axiom for real and complex numbers, justified by theorem axpre-mulext 7360 (Contributed by Jim Kingdon, 18-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) )