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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addgt0sr 11101 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
β’ ((0R <R π΄ β§ 0R <R π΅) β 0R <R (π΄ +R π΅)) | ||
Theorem | mulgt0sr 11102 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
β’ ((0R <R π΄ β§ 0R <R π΅) β 0R <R (π΄ Β·R π΅)) | ||
Theorem | sqgt0sr 11103 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β R β§ π΄ β 0R) β 0R <R (π΄ Β·R π΄)) | ||
Theorem | recexsr 11104* | The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
β’ ((π΄ β R β§ π΄ β 0R) β βπ₯ β R (π΄ Β·R π₯) = 1R) | ||
Theorem | mappsrpr 11105 | Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ πΆ β R β β’ ((πΆ +R -1R) <R (πΆ +R [β¨π΄, 1Pβ©] ~R ) β π΄ β P) | ||
Theorem | ltpsrpr 11106 | Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ πΆ β R β β’ ((πΆ +R [β¨π΄, 1Pβ©] ~R ) <R (πΆ +R [β¨π΅, 1Pβ©] ~R ) β π΄<P π΅) | ||
Theorem | map2psrpr 11107* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ πΆ β R β β’ ((πΆ +R -1R) <R π΄ β βπ₯ β P (πΆ +R [β¨π₯, 1Pβ©] ~R ) = π΄) | ||
Theorem | supsrlem 11108* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ π΅ = {π€ β£ (πΆ +R [β¨π€, 1Pβ©] ~R ) β π΄} & β’ πΆ β R β β’ ((πΆ β π΄ β§ βπ₯ β R βπ¦ β π΄ π¦ <R π₯) β βπ₯ β R (βπ¦ β π΄ Β¬ π₯ <R π¦ β§ βπ¦ β R (π¦ <R π₯ β βπ§ β π΄ π¦ <R π§))) | ||
Theorem | supsr 11109* | A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
β’ ((π΄ β β β§ βπ₯ β R βπ¦ β π΄ π¦ <R π₯) β βπ₯ β R (βπ¦ β π΄ Β¬ π₯ <R π¦ β§ βπ¦ β R (π¦ <R π₯ β βπ§ β π΄ π¦ <R π§))) | ||
Syntax | cc 11110 | Class of complex numbers. |
class β | ||
Syntax | cr 11111 | Class of real numbers. |
class β | ||
Syntax | cc0 11112 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 11113 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 11114 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 11115 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 11116 | 'Less than' predicate (defined over real subset of complex numbers). |
class <β | ||
Syntax | cmul 11117 | Multiplication on complex numbers. The token Β· is a center dot. |
class Β· | ||
Definition | df-c 11118 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 11145. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ β = (R Γ R) | ||
Definition | df-0 11119 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ 0 = β¨0R, 0Rβ© | ||
Definition | df-1 11120 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ 1 = β¨1R, 0Rβ© | ||
Definition | df-i 11121 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ i = β¨0R, 1Rβ© | ||
Definition | df-r 11122 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ β = (R Γ {0R}) | ||
Definition | df-add 11123* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
β’ + = {β¨β¨π₯, π¦β©, π§β© β£ ((π₯ β β β§ π¦ β β) β§ βπ€βπ£βπ’βπ((π₯ = β¨π€, π£β© β§ π¦ = β¨π’, πβ©) β§ π§ = β¨(π€ +R π’), (π£ +R π)β©))} | ||
Definition | df-mul 11124* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
β’ Β· = {β¨β¨π₯, π¦β©, π§β© β£ ((π₯ β β β§ π¦ β β) β§ βπ€βπ£βπ’βπ((π₯ = β¨π€, π£β© β§ π¦ = β¨π’, πβ©) β§ π§ = β¨((π€ Β·R π’) +R (-1R Β·R (π£ Β·R π))), ((π£ Β·R π’) +R (π€ Β·R π))β©))} | ||
Definition | df-lt 11125* | Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 11255. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ <β = {β¨π₯, π¦β© β£ ((π₯ β β β§ π¦ β β) β§ βπ§βπ€((π₯ = β¨π§, 0Rβ© β§ π¦ = β¨π€, 0Rβ©) β§ π§ <R π€))} | ||
Theorem | opelcn 11126 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
β’ (β¨π΄, π΅β© β β β (π΄ β R β§ π΅ β R)) | ||
Theorem | opelreal 11127 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
β’ (β¨π΄, 0Rβ© β β β π΄ β R) | ||
Theorem | elreal 11128* | 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 11129 | 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 11130 | 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 11131 | '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 11132 | 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 11133 | 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 11134 | 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 11135 | 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 11136 | 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 11137 | 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 11138 | 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 11139 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in β from those in R. The trick involves qsid 8779, 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 11118), 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 11140 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11139 and mulcnsrec 11141. (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 11141 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 8778,
which shows that the coset of
the converse membership relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 11139.
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 10841. (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 11142 | 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 11148. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11191. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
β’ + :(β Γ β)βΆβ | ||
Theorem | axmulf 11143 | 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 11150. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 11192. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
β’ Β· :(β Γ β)βΆβ | ||
Theorem | axcnex 11144 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12972), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5285 in later theorems by invoking Axiom ax-cnex 11168 instead of cnexALT 12972. Use cnex 11193 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
β’ β β V | ||
Theorem | axresscn 11145 | 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 11169. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
β’ β β β | ||
Theorem | ax1cn 11146 | 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 11170. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
β’ 1 β β | ||
Theorem | axicn 11147 | 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 11171. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
β’ i β β | ||
Theorem | axaddcl 11148 | 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 11172 be used later. Instead, in most cases use addcl 11194. (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 11149 | 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 11173 be used later. Instead, in most cases use readdcl 11195. (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 11150 | 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 11174 be used later. Instead, in most cases use mulcl 11196. (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 11151 | 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 11175 be used later. Instead, in most cases use remulcl 11197. (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 11152 | 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 11176 be used later. Instead, use mulcom 11198. (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 11153 | 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 11177 be used later. Instead, use addass 11199. (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 11154 | 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 11178. (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 11155 | 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 11179 be used later. Instead, use adddi 11201. (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 11156 | 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 11180. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
β’ ((i Β· i) + 1) = 0 | ||
Theorem | ax1ne0 11157 | 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 11181. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
β’ 1 β 0 | ||
Theorem | ax1rid 11158 | 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 mulrid 11214, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11182. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Theorem | axrnegex 11159* | 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 11183. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
β’ (π΄ β β β βπ₯ β β (π΄ + π₯) = 0) | ||
Theorem | axrrecex 11160* | 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 11184. (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 11161* | 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 11185. (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 11162 | 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 11287. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11186. (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 11163 | 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 11288. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11187. (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 11164 | 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 11289. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11188. (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 11165 | 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 11290. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11189. (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 11166* | 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 11291. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11190. (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 11167 | 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 11168 | The complex numbers form a set. This axiom is redundant - see cnexALT 12972- but we provide this axiom because the justification theorem axcnex 11144 does not use ax-rep 5285 even though the redundancy proof does. Proofs should normally use cnex 11193 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
β’ β β V | ||
Axiom | ax-resscn 11169 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by Theorem axresscn 11145. (Contributed by NM, 1-Mar-1995.) |
β’ β β β | ||
Axiom | ax-1cn 11170 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by Theorem ax1cn 11146. (Contributed by NM, 1-Mar-1995.) |
β’ 1 β β | ||
Axiom | ax-icn 11171 | i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by Theorem axicn 11147. (Contributed by NM, 1-Mar-1995.) |
β’ i β β | ||
Axiom | ax-addcl 11172 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11148. Proofs should normally use addcl 11194 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 11173 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl 11149. Proofs should normally use readdcl 11195 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 11174 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl 11150. Proofs should normally use mulcl 11196 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 11175 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl 11151. Proofs should normally use remulcl 11197 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 11176 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom 11152. Proofs should normally use mulcom 11198 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 11177 | Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass 11153. Proofs should normally use addass 11199 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 11178 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11154. Proofs should normally use mulass 11200 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 11179 | Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 11155. Proofs should normally use adddi 11201 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 11180 | 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 11156. (Contributed by NM, 29-Jan-1995.) |
β’ ((i Β· i) + 1) = 0 | ||
Axiom | ax-1ne0 11181 | 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by Theorem ax1ne0 11157. (Contributed by NM, 29-Jan-1995.) |
β’ 1 β 0 | ||
Axiom | ax-1rid 11182 | 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ax1rid 11158. Weakened from the original axiom in the form of statement in mulrid 11214, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) |
β’ (π΄ β β β (π΄ Β· 1) = π΄) | ||
Axiom | ax-rnegex 11183* | Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11159. (Contributed by Eric Schmidt, 21-May-2007.) |
β’ (π΄ β β β βπ₯ β β (π΄ + π₯) = 0) | ||
Axiom | ax-rrecex 11184* | Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11160. (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 11185* | 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 11161. For naming consistency, use cnre 11213 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 11186 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11162. Note: The more general version for extended reals is axlttri 11287. Normally new proofs would use xrlttri 13120. (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 11187 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11163. Note: The more general version for extended reals is axlttrn 11288. Normally new proofs would use lttr 11292. (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 11188 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11164. Normally new proofs would use axltadd 11289. (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 11189 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11165. Normally new proofs would use axmulgt0 11290. (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 11190* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11166. Note: Normally new proofs would use axsup 11291. (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 11191 |
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 11194 should be used. Note that uses of ax-addf 11191 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 11142. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ + :(β Γ β)βΆβ | ||
Axiom | ax-mulf 11192 |
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 11174 should be used. Note that uses of ax-mulf 11192
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 11143. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
β’ Β· :(β Γ β)βΆβ | ||
Theorem | cnex 11193 | Alias for ax-cnex 11168. See also cnexALT 12972. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ β β V | ||
Theorem | addcl 11194 | Alias for ax-addcl 11172, for naming consistency with addcli 11222. Use this theorem instead of ax-addcl 11172 or axaddcl 11148. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | readdcl 11195 | Alias for ax-addrcl 11173, for naming consistency with readdcli 11231. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) β β) | ||
Theorem | mulcl 11196 | Alias for ax-mulcl 11174, for naming consistency with mulcli 11223. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | remulcl 11197 | Alias for ax-mulrcl 11175, for naming consistency with remulcli 11232. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) β β) | ||
Theorem | mulcom 11198 | Alias for ax-mulcom 11176, for naming consistency with mulcomi 11224. (Contributed by NM, 10-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | addass 11199 | Alias for ax-addass 11177, for naming consistency with addassi 11226. (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 11200 | Alias for ax-mulass 11178, for naming consistency with mulassi 11227. (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))) |
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