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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elreal 11001* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
β’ (π΄ β β β βπ₯ β R β¨π₯, 0Rβ© = π΄) | ||
Theorem | elreal2 11002 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ (π΄ β β β ((1st βπ΄) β R β§ π΄ = β¨(1st βπ΄), 0Rβ©)) | ||
Theorem | 0ncn 11003 | 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.) (New usage is discouraged.) |
β’ Β¬ β β β | ||
Theorem | ltrelre 11004 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ <β β (β Γ β) | ||
Theorem | addcnsr 11005 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
β’ (((π΄ β R β§ π΅ β R) β§ (πΆ β R β§ π· β R)) β (β¨π΄, π΅β© + β¨πΆ, π·β©) = β¨(π΄ +R πΆ), (π΅ +R π·)β©) | ||
Theorem | mulcnsr 11006 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
β’ (((π΄ β R β§ π΅ β R) β§ (πΆ β R β§ π· β R)) β (β¨π΄, π΅β© Β· β¨πΆ, π·β©) = β¨((π΄ Β·R πΆ) +R (-1R Β·R (π΅ Β·R π·))), ((π΅ Β·R πΆ) +R (π΄ Β·R π·))β©) | ||
Theorem | eqresr 11007 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
β’ π΄ β V β β’ (β¨π΄, 0Rβ© = β¨π΅, 0Rβ© β π΄ = π΅) | ||
Theorem | addresr 11008 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β R β§ π΅ β R) β (β¨π΄, 0Rβ© + β¨π΅, 0Rβ©) = β¨(π΄ +R π΅), 0Rβ©) | ||
Theorem | mulresr 11009 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β R β§ π΅ β R) β (β¨π΄, 0Rβ© Β· β¨π΅, 0Rβ©) = β¨(π΄ Β·R π΅), 0Rβ©) | ||
Theorem | ltresr 11010 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ (β¨π΄, 0Rβ© <β β¨π΅, 0Rβ© β π΄ <R π΅) | ||
Theorem | ltresr2 11011 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ <β π΅ β (1st βπ΄) <R (1st βπ΅))) | ||
Theorem | dfcnqs 11012 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in β from those in R. The trick involves qsid 8656, which shows that the coset of the converse membership relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that β is a quotient set, even though it is not (compare df-c 10991), and allows 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.) (New usage is discouraged.) |
β’ β = ((R Γ R) / β‘ E ) | ||
Theorem | addcnsrec 11013 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11012 and mulcnsrec 11014. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
β’ (((π΄ β R β§ π΅ β R) β§ (πΆ β R β§ π· β R)) β ([β¨π΄, π΅β©]β‘ E + [β¨πΆ, π·β©]β‘ E ) = [β¨(π΄ +R πΆ), (π΅ +R π·)β©]β‘ E ) | ||
Theorem | mulcnsrec 11014 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 8655,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 11012.
Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 10714. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
β’ (((π΄ β R β§ π΅ β R) β§ (πΆ β R β§ π· β R)) β ([β¨π΄, π΅β©]β‘ E Β· [β¨πΆ, π·β©]β‘ E ) = [β¨((π΄ Β·R πΆ) +R (-1R Β·R (π΅ Β·R π·))), ((π΅ Β·R πΆ) +R (π΄ Β·R π·))β©]β‘ E ) | ||
Theorem | axaddf 11015 | 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 11021. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11064. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
β’ + :(β Γ β)βΆβ | ||
Theorem | axmulf 11016 | Multiplication 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 axmulcl 11023. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 11065. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
β’ Β· :(β Γ β)βΆβ | ||
Theorem | axcnex 11017 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12840), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5241 in later theorems by invoking Axiom ax-cnex 11041 instead of cnexALT 12840. Use cnex 11066 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
β’ β β V | ||
Theorem | axresscn 11018 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 11042. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
β’ β β β | ||
Theorem | ax1cn 11019 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 11043. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
β’ 1 β β | ||
Theorem | axicn 11020 | i is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 11044. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
β’ i β β | ||
Theorem | axaddcl 11021 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 11045 be used later. Instead, in most cases use addcl 11067. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | axaddrcl 11022 | Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 11046 be used later. Instead, in most cases use readdcl 11068. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | axmulcl 11023 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 11047 be used later. Instead, in most cases use mulcl 11069. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | axmulrcl 11024 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 11048 be used later. Instead, in most cases use remulcl 11070. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | axmulcom 11025 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 11049 be used later. Instead, use mulcom 11071. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | axaddass 11026 | 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 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 11050 be used later. Instead, use addass 11072. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) + πΆ) = (π΄ + (π΅ + πΆ))) | ||
Theorem | axmulass 11027 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 11051. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ Β· π΅) Β· πΆ) = (π΄ Β· (π΅ Β· πΆ))) | ||
Theorem | axdistr 11028 | Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 11052 be used later. Instead, use adddi 11074. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ Β· (π΅ + πΆ)) = ((π΄ Β· π΅) + (π΄ Β· πΆ))) | ||
Theorem | axi2m1 11029 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 11053. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
β’ ((i Β· i) + 1) = 0 | ||
Theorem | ax1ne0 11030 | 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 11054. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
β’ 1 β 0 | ||
Theorem | ax1rid 11031 | 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 11087, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11055. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Theorem | axrnegex 11032* | Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 11056. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
β’ (π΄ β β β βπ₯ β β (π΄ + π₯) = 0) | ||
Theorem | axrrecex 11033* | Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 11057. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΄ β 0) β βπ₯ β β (π΄ Β· π₯) = 1) | ||
Theorem | axcnre 11034* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 11058. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
β’ (π΄ β β β βπ₯ β β βπ¦ β β π΄ = (π₯ + (i Β· π¦))) | ||
Theorem | axpre-lttri 11035 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 11160. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11059. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ <β π΅ β Β¬ (π΄ = π΅ β¨ π΅ <β π΄))) | ||
Theorem | axpre-lttrn 11036 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 11161. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11060. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ <β π΅ β§ π΅ <β πΆ) β π΄ <β πΆ)) | ||
Theorem | axpre-ltadd 11037 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 11162. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11061. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΄ <β π΅ β (πΆ + π΄) <β (πΆ + π΅))) | ||
Theorem | axpre-mulgt0 11038 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 11163. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11062. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΅ β β) β ((0 <β π΄ β§ 0 <β π΅) β 0 <β (π΄ Β· π΅))) | ||
Theorem | axpre-sup 11039* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 11164. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11063. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
β’ ((π΄ β β β§ π΄ β β β§ βπ₯ β 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§))) | ||
Theorem | wuncn 11040 | A weak universe containing Ο contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ (π β π β WUni) & β’ (π β Ο β π) β β’ (π β β β π) | ||
Axiom | ax-cnex 11041 | The complex numbers form a set. This axiom is redundant - see cnexALT 12840- but we provide this axiom because the justification theorem axcnex 11017 does not use ax-rep 5241 even though the redundancy proof does. Proofs should normally use cnex 11066 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
β’ β β V | ||
Axiom | ax-resscn 11042 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by Theorem axresscn 11018. (Contributed by NM, 1-Mar-1995.) |
β’ β β β | ||
Axiom | ax-1cn 11043 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by Theorem ax1cn 11019. (Contributed by NM, 1-Mar-1995.) |
β’ 1 β β | ||
Axiom | ax-icn 11044 | i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by Theorem axicn 11020. (Contributed by NM, 1-Mar-1995.) |
β’ i β β | ||
Axiom | ax-addcl 11045 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11021. Proofs should normally use addcl 11067 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 11046 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl 11022. Proofs should normally use readdcl 11068 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 11047 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl 11023. Proofs should normally use mulcl 11069 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 11048 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl 11024. Proofs should normally use remulcl 11070 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-mulcom 11049 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom 11025. Proofs should normally use mulcom 11071 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 11050 | Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass 11026. Proofs should normally use addass 11072 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 11051 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11027. Proofs should normally use mulass 11073 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 11052 | Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 11028. Proofs should normally use adddi 11074 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 11053 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by Theorem axi2m1 11029. (Contributed by NM, 29-Jan-1995.) |
β’ ((i Β· i) + 1) = 0 | ||
Axiom | ax-1ne0 11054 | 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by Theorem ax1ne0 11030. (Contributed by NM, 29-Jan-1995.) |
β’ 1 β 0 | ||
Axiom | ax-1rid 11055 | 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ax1rid 11031. Weakened from the original axiom in the form of statement in mulid1 11087, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Axiom | ax-rnegex 11056* | Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11032. (Contributed by Eric Schmidt, 21-May-2007.) |
β’ (π΄ β β β βπ₯ β β (π΄ + π₯) = 0) | ||
Axiom | ax-rrecex 11057* | Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11033. (Contributed by Eric Schmidt, 11-Apr-2007.) |
β’ ((π΄ β β β§ π΄ β 0) β βπ₯ β β (π΄ Β· π₯) = 1) | ||
Axiom | ax-cnre 11058* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by Theorem axcnre 11034. For naming consistency, use cnre 11086 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-lttri 11059 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11035. Note: The more general version for extended reals is axlttri 11160. Normally new proofs would use xrlttri 12987. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ <β π΅ β Β¬ (π΄ = π΅ β¨ π΅ <β π΄))) | ||
Axiom | ax-pre-lttrn 11060 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11036. Note: The more general version for extended reals is axlttrn 11161. Normally new proofs would use lttr 11165. (New usage is discouraged.) (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-ltadd 11061 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11037. Normally new proofs would use axltadd 11162. (New usage is discouraged.) (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 11062 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11038. Normally new proofs would use axmulgt0 11163. (New usage is discouraged.) (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-sup 11063* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11039. Note: Normally new proofs would use axsup 11164. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
β’ ((π΄ β β β§ π΄ β β β§ βπ₯ β 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 11064 |
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-order 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 11067 should be used. Note that uses of ax-addf 11064 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 11015. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ + :(β Γ β)βΆβ | ||
Axiom | ax-mulf 11065 |
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-order 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 11047 should be used. Note that uses of ax-mulf 11065
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 11016. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ Β· :(β Γ β)βΆβ | ||
Theorem | cnex 11066 | Alias for ax-cnex 11041. See also cnexALT 12840. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ β β V | ||
Theorem | addcl 11067 | Alias for ax-addcl 11045, for naming consistency with addcli 11095. Use this theorem instead of ax-addcl 11045 or axaddcl 11021. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | readdcl 11068 | Alias for ax-addrcl 11046, for naming consistency with readdcli 11104. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | mulcl 11069 | Alias for ax-mulcl 11047, for naming consistency with mulcli 11096. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | remulcl 11070 | Alias for ax-mulrcl 11048, for naming consistency with remulcli 11105. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | mulcom 11071 | Alias for ax-mulcom 11049, for naming consistency with mulcomi 11097. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | addass 11072 | Alias for ax-addass 11050, for naming consistency with addassi 11099. (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 11073 | Alias for ax-mulass 11051, for naming consistency with mulassi 11100. (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 11074 | Alias for ax-distr 11052, for naming consistency with adddii 11101. (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 11075 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
β’ (π΄ β β β π΄ β β) | ||
Theorem | reex 11076 | The real numbers form a set. See also reexALT 12838. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ β β V | ||
Theorem | reelprrecn 11077 | Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ β β {β, β} | ||
Theorem | cnelprrecn 11078 | Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ β β {β, β} | ||
Theorem | elimne0 11079 | Hypothesis for weak deduction theorem to eliminate π΄ β 0. (Contributed by NM, 15-May-1999.) |
β’ if(π΄ β 0, π΄, 1) β 0 | ||
Theorem | adddir 11080 | 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 11081 | Zero is a complex number. See also 0cnALT 11323. (Contributed by NM, 19-Feb-2005.) |
β’ 0 β β | ||
Theorem | 0cnd 11082 | Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
β’ (π β 0 β β) | ||
Theorem | c0ex 11083 | Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.) |
β’ 0 β V | ||
Theorem | 1cnd 11084 | One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 1 β β) | ||
Theorem | 1ex 11085 | One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.) |
β’ 1 β V | ||
Theorem | cnre 11086* | Alias for ax-cnre 11058, 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 | mulid1 11087 | The number 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 | mulid2 11088 | Identity law for multiplication. See mulid1 11087 for commuted version. (Contributed by NM, 8-Oct-1999.) |
β’ (π΄ β β β (1 Β· π΄) = π΄) | ||
Theorem | 1re 11089 | The number 1 is real. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 11043, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
β’ 1 β β | ||
Theorem | 1red 11090 | The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 1 β β) | ||
Theorem | 0re 11091 | The number 0 is real. Remark: the first step could also be ax-icn 11044. See also 0reALT 11432. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 11-Oct-2022.) |
β’ 0 β β | ||
Theorem | 0red 11092 | The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
β’ (π β 0 β β) | ||
Theorem | mulid1i 11093 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
β’ π΄ β β β β’ (π΄ Β· 1) = π΄ | ||
Theorem | mulid2i 11094 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
β’ π΄ β β β β’ (1 Β· π΄) = π΄ | ||
Theorem | addcli 11095 | 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 11096 | 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 11097 | 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 11098 | 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 11099 | 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 11100 | 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)) |
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