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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rereceu 7901* | The reciprocal from axprecex 7892 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.) |
β’ ((π΄ β β β§ 0 <β π΄) β β!π₯ β β (π΄ Β· π₯) = 1) | ||
Theorem | recriota 7902* | Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.) |
β’ (π β N β (β©π β β (β¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ© Β· π) = 1) = β¨[β¨(β¨{π β£ π <Q (*Qβ[β¨π, 1oβ©] ~Q )}, {π’ β£ (*Qβ[β¨π, 1oβ©] ~Q ) <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) | ||
Theorem | axarch 7903* |
Archimedean axiom. The Archimedean property is more naturally stated
once we have defined β. Unless we find
another way to state it,
we'll just use the right hand side of dfnn2 8934 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 7943. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.) |
β’ (π΄ β β β βπ β β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)}π΄ <β π) | ||
Theorem | peano5nnnn 7904* | Peano's inductive postulate. This is a counterpart to peano5nni 8935 designed for real number axioms which involve natural numbers (notably, axcaucvg 7912). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} β β’ ((1 β π΄ β§ βπ§ β π΄ (π§ + 1) β π΄) β π β π΄) | ||
Theorem | nnindnn 7905* | Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8948 designed for real number axioms which involve natural numbers (notably, axcaucvg 7912). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π§ = 1 β (π β π)) & β’ (π§ = π β (π β π)) & β’ (π§ = (π + 1) β (π β π)) & β’ (π§ = π΄ β (π β π)) & β’ π & β’ (π β π β (π β π)) β β’ (π΄ β π β π) | ||
Theorem | nntopi 7906* | Mapping from β to N. (Contributed by Jim Kingdon, 13-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} β β’ (π΄ β π β βπ§ β N β¨[β¨(β¨{π β£ π <Q [β¨π§, 1oβ©] ~Q }, {π’ β£ [β¨π§, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ© = π΄) | ||
Theorem | axcaucvglemcl 7907* | Lemma for axcaucvg 7912. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) β β’ ((π β§ π½ β N) β (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π½, 1oβ©] ~Q }, {π’ β£ [β¨π½, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©) β R) | ||
Theorem | axcaucvglemf 7908* | Lemma for axcaucvg 7912. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) & β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) β β’ (π β πΊ:NβΆR) | ||
Theorem | axcaucvglemval 7909* | Lemma for axcaucvg 7912. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) & β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) β β’ ((π β§ π½ β N) β (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π½, 1oβ©] ~Q }, {π’ β£ [β¨π½, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨(πΊβπ½), 0Rβ©) | ||
Theorem | axcaucvglemcau 7910* | Lemma for axcaucvg 7912. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) & β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) β β’ (π β βπ β N βπ β N (π <N π β ((πΊβπ) <R ((πΊβπ) +R [β¨(β¨{π β£ π <Q (*Qβ[β¨π, 1oβ©] ~Q )}, {π’ β£ (*Qβ[β¨π, 1oβ©] ~Q ) <Q π’}β© +P 1P), 1Pβ©] ~R ) β§ (πΊβπ) <R ((πΊβπ) +R [β¨(β¨{π β£ π <Q (*Qβ[β¨π, 1oβ©] ~Q )}, {π’ β£ (*Qβ[β¨π, 1oβ©] ~Q ) <Q π’}β© +P 1P), 1Pβ©] ~R )))) | ||
Theorem | axcaucvglemres 7911* | Lemma for axcaucvg 7912. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) & β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) β β’ (π β βπ¦ β β βπ₯ β β (0 <β π₯ β βπ β π βπ β π (π <β π β ((πΉβπ) <β (π¦ + π₯) β§ π¦ <β ((πΉβπ) + π₯))))) | ||
Theorem | axcaucvg 7912* |
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 / π 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 β or division, we use π for the natural numbers and express a reciprocal in terms of β©. This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7944. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) β β’ (π β βπ¦ β β βπ₯ β β (0 <β π₯ β βπ β π βπ β π (π <β π β ((πΉβπ) <β (π¦ + π₯) β§ π¦ <β ((πΉβπ) + π₯))))) | ||
Theorem | axpre-suploclemres 7913* | Lemma for axpre-suploc 7914. The result. The proof just needs to define π΅ as basically the same set as π΄ (but expressed as a subset of R rather than a subset of β), and apply suplocsr 7821. (Contributed by Jim Kingdon, 24-Jan-2024.) |
β’ (π β π΄ β β) & β’ (π β πΆ β π΄) & β’ (π β βπ₯ β β βπ¦ β π΄ π¦ <β π₯) & β’ (π β βπ₯ β β βπ¦ β β (π₯ <β π¦ β (βπ§ β π΄ π₯ <β π§ β¨ βπ§ β π΄ π§ <β π¦))) & β’ π΅ = {π€ β R β£ β¨π€, 0Rβ© β π΄} β β’ (π β βπ₯ β β (βπ¦ β π΄ Β¬ π₯ <β π¦ β§ βπ¦ β β (π¦ <β π₯ β βπ§ β π΄ π¦ <β π§))) | ||
Theorem | axpre-suploc 7914* |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given π₯ < π¦, either there is an element of the set greater than π₯, or π¦ is an upper bound. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7945. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.) |
β’ (((π΄ β β β§ βπ₯ π₯ β π΄) β§ (βπ₯ β β βπ¦ β π΄ π¦ <β π₯ β§ βπ₯ β β βπ¦ β β (π₯ <β π¦ β (βπ§ β π΄ π₯ <β π§ β¨ βπ§ β π΄ π§ <β π¦)))) β βπ₯ β β (βπ¦ β π΄ Β¬ π₯ <β π¦ β§ βπ¦ β β (π¦ <β π₯ β βπ§ β π΄ π¦ <β π§))) | ||
Axiom | ax-cnex 7915 | The complex numbers form a set. Proofs should normally use cnex 7948 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
β’ β β V | ||
Axiom | ax-resscn 7916 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 7872. (Contributed by NM, 1-Mar-1995.) |
β’ β β β | ||
Axiom | ax-1cn 7917 | 1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 7873. (Contributed by NM, 1-Mar-1995.) |
β’ 1 β β | ||
Axiom | ax-1re 7918 | 1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 7874. Proofs should use 1re 7969 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
β’ 1 β β | ||
Axiom | ax-icn 7919 | i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 7875. (Contributed by NM, 1-Mar-1995.) |
β’ i β β | ||
Axiom | ax-addcl 7920 | Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 7876. Proofs should normally use addcl 7949 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Axiom | ax-addrcl 7921 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 7877. Proofs should normally use readdcl 7950 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Axiom | ax-mulcl 7922 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7878. Proofs should normally use mulcl 7951 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Axiom | ax-mulrcl 7923 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7879. Proofs should normally use remulcl 7952 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Axiom | ax-addcom 7924 | Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 7882. Proofs should normally use addcom 8107 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) = (π΅ + π΄)) | ||
Axiom | ax-mulcom 7925 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 7883. Proofs should normally use mulcom 7953 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Axiom | ax-addass 7926 | Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7884. Proofs should normally use addass 7954 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) + πΆ) = (π΄ + (π΅ + πΆ))) | ||
Axiom | ax-mulass 7927 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7885. Proofs should normally use mulass 7955 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ Β· π΅) Β· πΆ) = (π΄ Β· (π΅ Β· πΆ))) | ||
Axiom | ax-distr 7928 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 7886. Proofs should normally use adddi 7956 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ Β· (π΅ + πΆ)) = ((π΄ Β· π΅) + (π΄ Β· πΆ))) | ||
Axiom | ax-i2m1 7929 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 7887. (Contributed by NM, 29-Jan-1995.) |
β’ ((i Β· i) + 1) = 0 | ||
Axiom | ax-0lt1 7930 | 0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 7888. Proofs should normally use 0lt1 8097 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) |
β’ 0 <β 1 | ||
Axiom | ax-1rid 7931 | 1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 7889. (Contributed by NM, 29-Jan-1995.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Axiom | ax-0id 7932 |
0 is an identity element for real addition. Axiom for
real and
complex numbers, justified by Theorem ax0id 7890.
Proofs should normally use addid1 8108 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) |
β’ (π΄ β β β (π΄ + 0) = π΄) | ||
Axiom | ax-rnegex 7933* | Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 7891. (Contributed by Eric Schmidt, 21-May-2007.) |
β’ (π΄ β β β βπ₯ β β (π΄ + π₯) = 0) | ||
Axiom | ax-precex 7934* | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 7892. (Contributed by Jim Kingdon, 6-Feb-2020.) |
β’ ((π΄ β β β§ 0 <β π΄) β βπ₯ β β (0 <β π₯ β§ (π΄ Β· π₯) = 1)) | ||
Axiom | ax-cnre 7935* | 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 7893. For naming consistency, use cnre 7966 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
β’ (π΄ β β β βπ₯ β β βπ¦ β β π΄ = (π₯ + (i Β· π¦))) | ||
Axiom | ax-pre-ltirr 7936 | Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 7936. (Contributed by Jim Kingdon, 12-Jan-2020.) |
β’ (π΄ β β β Β¬ π΄ <β π΄) | ||
Axiom | ax-pre-ltwlin 7937 | Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7895. (Contributed by Jim Kingdon, 12-Jan-2020.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ <β π΅ β (π΄ <β πΆ β¨ πΆ <β π΅))) | ||
Axiom | ax-pre-lttrn 7938 | Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 7896. (Contributed by NM, 13-Oct-2005.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ <β π΅ β§ π΅ <β πΆ) β π΄ <β πΆ)) | ||
Axiom | ax-pre-apti 7939 | Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7897. (Contributed by Jim Kingdon, 29-Jan-2020.) |
β’ ((π΄ β β β§ π΅ β β β§ Β¬ (π΄ <β π΅ β¨ π΅ <β π΄)) β π΄ = π΅) | ||
Axiom | ax-pre-ltadd 7940 | Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 7898. (Contributed by NM, 13-Oct-2005.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ <β π΅ β (πΆ + π΄) <β (πΆ + π΅))) | ||
Axiom | ax-pre-mulgt0 7941 | The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 7899. (Contributed by NM, 13-Oct-2005.) |
β’ ((π΄ β β β§ π΅ β β) β ((0 <β π΄ β§ 0 <β π΅) β 0 <β (π΄ Β· π΅))) | ||
Axiom | ax-pre-mulext 7942 |
Strong extensionality of multiplication (expressed in terms of <β).
Axiom for real and complex numbers, justified by Theorem axpre-mulext 7900
(Contributed by Jim Kingdon, 18-Feb-2020.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ Β· πΆ) <β (π΅ Β· πΆ) β (π΄ <β π΅ β¨ π΅ <β π΄))) | ||
Axiom | ax-arch 7943* |
Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
real and complex numbers, justified by Theorem axarch 7903.
This axiom should not be used directly; instead use arch 9186 (which is the same, but stated in terms of β and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.) |
β’ (π΄ β β β βπ β β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)}π΄ <β π) | ||
Axiom | ax-caucvg 7944* |
Completeness. Axiom for real and complex numbers, justified by Theorem
axcaucvg 7912.
A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / π of the nth term. This axiom should not be used directly; instead use caucvgre 11003 (which is the same, but stated in terms of the β and 1 / π notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.) |
β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} & β’ (π β πΉ:πβΆβ) & β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) β β’ (π β βπ¦ β β βπ₯ β β (0 <β π₯ β βπ β π βπ β π (π <β π β ((πΉβπ) <β (π¦ + π₯) β§ π¦ <β ((πΉβπ) + π₯))))) | ||
Axiom | ax-pre-suploc 7945* |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given π₯ < π¦, either there is an element of the set greater than π₯, or π¦ is an upper bound. Although this and ax-caucvg 7944 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7944. (Contributed by Jim Kingdon, 23-Jan-2024.) |
β’ (((π΄ β β β§ βπ₯ π₯ β π΄) β§ (βπ₯ β β βπ¦ β π΄ π¦ <β π₯ β§ βπ₯ β β βπ¦ β β (π₯ <β π¦ β (βπ§ β π΄ π₯ <β π§ β¨ βπ§ β π΄ π§ <β π¦)))) β βπ₯ β β (βπ¦ β π΄ Β¬ π₯ <β π¦ β§ βπ¦ β β (π¦ <β π₯ β βπ§ β π΄ π¦ <β π§))) | ||
Axiom | ax-addf 7946 |
Addition is an operation on the complex numbers. This deprecated axiom is
provided for historical compatibility but is not a bona fide axiom for
complex numbers (independent of set theory) since it cannot be interpreted
as a first- or second-order statement (see
https://us.metamath.org/downloads/schmidt-cnaxioms.pdf).
It may be
deleted in the future and should be avoided for new theorems. Instead,
the less specific addcl 7949 should be used. Note that uses of ax-addf 7946 can
be eliminated by using the defined operation
(π₯
β β, π¦ β
β β¦ (π₯ + π¦)) in place of +, from which
this axiom (with the defined operation in place of +) follows as a
theorem.
This axiom is justified by Theorem axaddf 7880. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ + :(β Γ β)βΆβ | ||
Axiom | ax-mulf 7947 |
Multiplication is an operation on the complex numbers. This deprecated
axiom is provided for historical compatibility but is not a bona fide
axiom for complex numbers (independent of set theory) since it cannot be
interpreted as a first- or second-order statement (see
https://us.metamath.org/downloads/schmidt-cnaxioms.pdf).
It may be
deleted in the future and should be avoided for new theorems. Instead,
the less specific ax-mulcl 7922 should be used. Note that uses of ax-mulf 7947
can be eliminated by using the defined operation
(π₯
β β, π¦ β
β β¦ (π₯ Β·
π¦)) in place of
Β·, from which
this axiom (with the defined operation in place of Β·) follows as a
theorem.
This axiom is justified by Theorem axmulf 7881. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ Β· :(β Γ β)βΆβ | ||
Theorem | cnex 7948 | Alias for ax-cnex 7915. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ β β V | ||
Theorem | addcl 7949 | Alias for ax-addcl 7920, for naming consistency with addcli 7974. Use this theorem instead of ax-addcl 7920 or axaddcl 7876. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | readdcl 7950 | Alias for ax-addrcl 7921, for naming consistency with readdcli 7983. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | mulcl 7951 | Alias for ax-mulcl 7922, for naming consistency with mulcli 7975. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | remulcl 7952 | Alias for ax-mulrcl 7923, for naming consistency with remulcli 7984. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | mulcom 7953 | Alias for ax-mulcom 7925, for naming consistency with mulcomi 7976. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | addass 7954 | Alias for ax-addass 7926, for naming consistency with addassi 7978. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) + πΆ) = (π΄ + (π΅ + πΆ))) | ||
Theorem | mulass 7955 | Alias for ax-mulass 7927, for naming consistency with mulassi 7979. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ Β· π΅) Β· πΆ) = (π΄ Β· (π΅ Β· πΆ))) | ||
Theorem | adddi 7956 | Alias for ax-distr 7928, for naming consistency with adddii 7980. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ Β· (π΅ + πΆ)) = ((π΄ Β· π΅) + (π΄ Β· πΆ))) | ||
Theorem | recn 7957 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
β’ (π΄ β β β π΄ β β) | ||
Theorem | reex 7958 | The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ β β V | ||
Theorem | reelprrecn 7959 | Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ β β {β, β} | ||
Theorem | cnelprrecn 7960 | Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ β β {β, β} | ||
Theorem | adddir 7961 | Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) Β· πΆ) = ((π΄ Β· πΆ) + (π΅ Β· πΆ))) | ||
Theorem | 0cn 7962 | 0 is a complex number. (Contributed by NM, 19-Feb-2005.) |
β’ 0 β β | ||
Theorem | 0cnd 7963 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ (π β 0 β β) | ||
Theorem | c0ex 7964 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
β’ 0 β V | ||
Theorem | 1ex 7965 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
β’ 1 β V | ||
Theorem | cnre 7966* | Alias for ax-cnre 7935, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
β’ (π΄ β β β βπ₯ β β βπ¦ β β π΄ = (π₯ + (i Β· π¦))) | ||
Theorem | mulrid 7967 | 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Theorem | mullid 7968 | Identity law for multiplication. Note: see mulrid 7967 for commuted version. (Contributed by NM, 8-Oct-1999.) |
β’ (π΄ β β β (1 Β· π΄) = π΄) | ||
Theorem | 1re 7969 | 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) |
β’ 1 β β | ||
Theorem | 0re 7970 | 0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
β’ 0 β β | ||
Theorem | 0red 7971 | 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 0 β β) | ||
Theorem | mulid1i 7972 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
β’ π΄ β β β β’ (π΄ Β· 1) = π΄ | ||
Theorem | mullidi 7973 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
β’ π΄ β β β β’ (1 Β· π΄) = π΄ | ||
Theorem | addcli 7974 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β β β’ (π΄ + π΅) β β | ||
Theorem | mulcli 7975 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β β β’ (π΄ Β· π΅) β β | ||
Theorem | mulcomi 7976 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β β β’ (π΄ Β· π΅) = (π΅ Β· π΄) | ||
Theorem | mulcomli 7977 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β & β’ (π΄ Β· π΅) = πΆ β β’ (π΅ Β· π΄) = πΆ | ||
Theorem | addassi 7978 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β & β’ πΆ β β β β’ ((π΄ + π΅) + πΆ) = (π΄ + (π΅ + πΆ)) | ||
Theorem | mulassi 7979 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β & β’ πΆ β β β β’ ((π΄ Β· π΅) Β· πΆ) = (π΄ Β· (π΅ Β· πΆ)) | ||
Theorem | adddii 7980 | Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
β’ π΄ β β & β’ π΅ β β & β’ πΆ β β β β’ (π΄ Β· (π΅ + πΆ)) = ((π΄ Β· π΅) + (π΄ Β· πΆ)) | ||
Theorem | adddiri 7981 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
β’ π΄ β β & β’ π΅ β β & β’ πΆ β β β β’ ((π΄ + π΅) Β· πΆ) = ((π΄ Β· πΆ) + (π΅ Β· πΆ)) | ||
Theorem | recni 7982 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
β’ π΄ β β β β’ π΄ β β | ||
Theorem | readdcli 7983 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
β’ π΄ β β & β’ π΅ β β β β’ (π΄ + π΅) β β | ||
Theorem | remulcli 7984 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
β’ π΄ β β & β’ π΅ β β β β’ (π΄ Β· π΅) β β | ||
Theorem | 1red 7985 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 1 β β) | ||
Theorem | 1cnd 7986 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 1 β β) | ||
Theorem | mulridd 7987 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (π΄ Β· 1) = π΄) | ||
Theorem | mullidd 7988 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (1 Β· π΄) = π΄) | ||
Theorem | mulid2d 7989 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (1 Β· π΄) = π΄) | ||
Theorem | addcld 7990 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (π΄ + π΅) β β) | ||
Theorem | mulcld 7991 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (π΄ Β· π΅) β β) | ||
Theorem | mulcomd 7992 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | addassd 7993 | Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) β β’ (π β ((π΄ + π΅) + πΆ) = (π΄ + (π΅ + πΆ))) | ||
Theorem | mulassd 7994 | Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) β β’ (π β ((π΄ Β· π΅) Β· πΆ) = (π΄ Β· (π΅ Β· πΆ))) | ||
Theorem | adddid 7995 | Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) β β’ (π β (π΄ Β· (π΅ + πΆ)) = ((π΄ Β· π΅) + (π΄ Β· πΆ))) | ||
Theorem | adddird 7996 | Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) β β’ (π β ((π΄ + π΅) Β· πΆ) = ((π΄ Β· πΆ) + (π΅ Β· πΆ))) | ||
Theorem | adddirp1d 7997 | Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β ((π΄ + 1) Β· π΅) = ((π΄ Β· π΅) + π΅)) | ||
Theorem | joinlmuladdmuld 7998 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β ((π΄ Β· π΅) + (πΆ Β· π΅)) = π·) β β’ (π β ((π΄ + πΆ) Β· π΅) = π·) | ||
Theorem | recnd 7999 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
β’ (π β π΄ β β) β β’ (π β π΄ β β) | ||
Theorem | readdcld 8000 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (π΄ + π΅) β β) |
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