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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelreal 11001* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ βˆƒπ‘₯ ∈ R ⟨π‘₯, 0R⟩ = 𝐴)
 
Theoremelreal2 11002 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ((1st β€˜π΄) ∈ R ∧ 𝐴 = ⟨(1st β€˜π΄), 0R⟩))
 
Theorem0ncn 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¬ βˆ… ∈ β„‚
 
Theoremltrelre 11004 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
<ℝ βŠ† (ℝ Γ— ℝ)
 
Theoremaddcnsr 11005 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
(((𝐴 ∈ R ∧ 𝐡 ∈ R) ∧ (𝐢 ∈ R ∧ 𝐷 ∈ R)) β†’ (⟨𝐴, 𝐡⟩ + ⟨𝐢, 𝐷⟩) = ⟨(𝐴 +R 𝐢), (𝐡 +R 𝐷)⟩)
 
Theoremmulcnsr 11006 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
(((𝐴 ∈ R ∧ 𝐡 ∈ R) ∧ (𝐢 ∈ R ∧ 𝐷 ∈ R)) β†’ (⟨𝐴, 𝐡⟩ Β· ⟨𝐢, 𝐷⟩) = ⟨((𝐴 Β·R 𝐢) +R (-1R Β·R (𝐡 Β·R 𝐷))), ((𝐡 Β·R 𝐢) +R (𝐴 Β·R 𝐷))⟩)
 
Theoremeqresr 11007 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
𝐴 ∈ V    β‡’   (⟨𝐴, 0R⟩ = ⟨𝐡, 0R⟩ ↔ 𝐴 = 𝐡)
 
Theoremaddresr 11008 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴 ∈ R ∧ 𝐡 ∈ R) β†’ (⟨𝐴, 0R⟩ + ⟨𝐡, 0R⟩) = ⟨(𝐴 +R 𝐡), 0R⟩)
 
Theoremmulresr 11009 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴 ∈ R ∧ 𝐡 ∈ R) β†’ (⟨𝐴, 0R⟩ Β· ⟨𝐡, 0R⟩) = ⟨(𝐴 Β·R 𝐡), 0R⟩)
 
Theoremltresr 11010 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ <ℝ ⟨𝐡, 0R⟩ ↔ 𝐴 <R 𝐡)
 
Theoremltresr2 11011 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 <ℝ 𝐡 ↔ (1st β€˜π΄) <R (1st β€˜π΅)))
 
Theoremdfcnqs 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„‚ = ((R Γ— R) / β—‘ E )
 
Theoremaddcnsrec 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.)
(((𝐴 ∈ R ∧ 𝐡 ∈ R) ∧ (𝐢 ∈ R ∧ 𝐷 ∈ R)) β†’ ([⟨𝐴, 𝐡⟩]β—‘ E + [⟨𝐢, 𝐷⟩]β—‘ E ) = [⟨(𝐴 +R 𝐢), (𝐡 +R 𝐷)⟩]β—‘ E )
 
Theoremmulcnsrec 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.)

(((𝐴 ∈ R ∧ 𝐡 ∈ R) ∧ (𝐢 ∈ R ∧ 𝐷 ∈ R)) β†’ ([⟨𝐴, 𝐡⟩]β—‘ E Β· [⟨𝐢, 𝐷⟩]β—‘ E ) = [⟨((𝐴 Β·R 𝐢) +R (-1R Β·R (𝐡 Β·R 𝐷))), ((𝐡 Β·R 𝐢) +R (𝐴 Β·R 𝐷))⟩]β—‘ E )
 
5.1.2  Final derivation of real and complex number postulates
 
Theoremaxaddf 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„‚ Γ— β„‚)βŸΆβ„‚
 
Theoremaxmulf 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„‚ Γ— β„‚)βŸΆβ„‚
 
Theoremaxcnex 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„‚ ∈ V
 
Theoremaxresscn 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.)
ℝ βŠ† β„‚
 
Theoremax1cn 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.)
1 ∈ β„‚
 
Theoremaxicn 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.)
i ∈ β„‚
 
Theoremaxaddcl 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 + 𝐡) ∈ β„‚)
 
Theoremaxaddrcl 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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Theoremaxmulcl 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Theoremaxmulrcl 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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Theoremaxmulcom 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Theoremaxaddass 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢)))
 
Theoremaxmulass 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 Β· 𝐡) Β· 𝐢) = (𝐴 Β· (𝐡 Β· 𝐢)))
 
Theoremaxdistr 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ (𝐴 Β· (𝐡 + 𝐢)) = ((𝐴 Β· 𝐡) + (𝐴 Β· 𝐢)))
 
Theoremaxi2m1 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.)
((i Β· i) + 1) = 0
 
Theoremax1ne0 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.)
1 β‰  0
 
Theoremax1rid 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.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· 1) = 𝐴)
 
Theoremaxrnegex 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.)
(𝐴 ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 + π‘₯) = 0)
 
Theoremaxrrecex 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.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 Β· π‘₯) = 1)
 
Theoremaxcnre 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.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯ + (i Β· 𝑦)))
 
Theoremaxpre-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 <ℝ 𝐡 ↔ Β¬ (𝐴 = 𝐡 ∨ 𝐡 <ℝ 𝐴)))
 
Theoremaxpre-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 <ℝ 𝐡 ∧ 𝐡 <ℝ 𝐢) β†’ 𝐴 <ℝ 𝐢))
 
Theoremaxpre-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ (𝐴 <ℝ 𝐡 β†’ (𝐢 + 𝐴) <ℝ (𝐢 + 𝐡)))
 
Theoremaxpre-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐡) β†’ 0 <ℝ (𝐴 Β· 𝐡)))
 
Theoremaxpre-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.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
Theoremwuncn 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.)
(πœ‘ β†’ π‘ˆ ∈ WUni)    &   (πœ‘ β†’ Ο‰ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ β„‚ ∈ π‘ˆ)
 
5.1.3  Real and complex number postulates restated as axioms
 
Axiomax-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„‚ ∈ V
 
Axiomax-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.)
ℝ βŠ† β„‚
 
Axiomax-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.)
1 ∈ β„‚
 
Axiomax-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.)
i ∈ β„‚
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 + 𝐡) ∈ β„‚)
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢)))
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 Β· 𝐡) Β· 𝐢) = (𝐴 Β· (𝐡 Β· 𝐢)))
 
Axiomax-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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ (𝐴 Β· (𝐡 + 𝐢)) = ((𝐴 Β· 𝐡) + (𝐴 Β· 𝐢)))
 
Axiomax-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.)
((i Β· i) + 1) = 0
 
Axiomax-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.)
1 β‰  0
 
Axiomax-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.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· 1) = 𝐴)
 
Axiomax-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.)
(𝐴 ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 + π‘₯) = 0)
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 Β· π‘₯) = 1)
 
Axiomax-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.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯ + (i Β· 𝑦)))
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 <ℝ 𝐡 ↔ Β¬ (𝐴 = 𝐡 ∨ 𝐡 <ℝ 𝐴)))
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 <ℝ 𝐡 ∧ 𝐡 <ℝ 𝐢) β†’ 𝐴 <ℝ 𝐢))
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ (𝐴 <ℝ 𝐡 β†’ (𝐢 + 𝐴) <ℝ (𝐢 + 𝐡)))
 
Axiomax-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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐡) β†’ 0 <ℝ (𝐴 Β· 𝐡)))
 
Axiomax-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.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
Axiomax-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„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ + 𝑦)) 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„‚ Γ— β„‚)βŸΆβ„‚
 
Axiomax-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„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) 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„‚ Γ— β„‚)βŸΆβ„‚
 
5.2  Derive the basic properties from the field axioms
 
5.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 11066 Alias for ax-cnex 11041. See also cnexALT 12840. (Contributed by Mario Carneiro, 17-Nov-2014.)
β„‚ ∈ V
 
Theoremaddcl 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 + 𝐡) ∈ β„‚)
 
Theoremreaddcl 11068 Alias for ax-addrcl 11046, for naming consistency with readdcli 11104. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Theoremmulcl 11069 Alias for ax-mulcl 11047, for naming consistency with mulcli 11096. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Theoremremulcl 11070 Alias for ax-mulrcl 11048, for naming consistency with remulcli 11105. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Theoremmulcom 11071 Alias for ax-mulcom 11049, for naming consistency with mulcomi 11097. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Theoremaddass 11072 Alias for ax-addass 11050, for naming consistency with addassi 11099. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢)))
 
Theoremmulass 11073 Alias for ax-mulass 11051, for naming consistency with mulassi 11100. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 Β· 𝐡) Β· 𝐢) = (𝐴 Β· (𝐡 Β· 𝐢)))
 
Theoremadddi 11074 Alias for ax-distr 11052, for naming consistency with adddii 11101. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ (𝐴 Β· (𝐡 + 𝐢)) = ((𝐴 Β· 𝐡) + (𝐴 Β· 𝐢)))
 
Theoremrecn 11075 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ β†’ 𝐴 ∈ β„‚)
 
Theoremreex 11076 The real numbers form a set. See also reexALT 12838. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V
 
Theoremreelprrecn 11077 Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.)
ℝ ∈ {ℝ, β„‚}
 
Theoremcnelprrecn 11078 Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.)
β„‚ ∈ {ℝ, β„‚}
 
Theoremelimne0 11079 Hypothesis for weak deduction theorem to eliminate 𝐴 β‰  0. (Contributed by NM, 15-May-1999.)
if(𝐴 β‰  0, 𝐴, 1) β‰  0
 
Theoremadddir 11080 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) Β· 𝐢) = ((𝐴 Β· 𝐢) + (𝐡 Β· 𝐢)))
 
Theorem0cn 11081 Zero is a complex number. See also 0cnALT 11323. (Contributed by NM, 19-Feb-2005.)
0 ∈ β„‚
 
Theorem0cnd 11082 Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(πœ‘ β†’ 0 ∈ β„‚)
 
Theoremc0ex 11083 Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V
 
Theorem1cnd 11084 One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(πœ‘ β†’ 1 ∈ β„‚)
 
Theorem1ex 11085 One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V
 
Theoremcnre 11086* Alias for ax-cnre 11058, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯ + (i Β· 𝑦)))
 
Theoremmulid1 11087 The number 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β· 1) = 𝐴)
 
Theoremmulid2 11088 Identity law for multiplication. See mulid1 11087 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ β„‚ β†’ (1 Β· 𝐴) = 𝐴)
 
Theorem1re 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.)
1 ∈ ℝ
 
Theorem1red 11090 The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(πœ‘ β†’ 1 ∈ ℝ)
 
Theorem0re 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.)
0 ∈ ℝ
 
Theorem0red 11092 The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(πœ‘ β†’ 0 ∈ ℝ)
 
Theoremmulid1i 11093 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ β„‚    β‡’   (𝐴 Β· 1) = 𝐴
 
Theoremmulid2i 11094 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ β„‚    β‡’   (1 Β· 𝐴) = 𝐴
 
Theoremaddcli 11095 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (𝐴 + 𝐡) ∈ β„‚
 
Theoremmulcli 11096 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (𝐴 Β· 𝐡) ∈ β„‚
 
Theoremmulcomi 11097 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴)
 
Theoremmulcomli 11098 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   (𝐴 Β· 𝐡) = 𝐢    β‡’   (𝐡 Β· 𝐴) = 𝐢
 
Theoremaddassi 11099 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‚    β‡’   ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢))
 
Theoremmulassi 11100 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‚    β‡’   ((𝐴 Β· 𝐡) Β· 𝐢) = (𝐴 Β· (𝐡 Β· 𝐢))
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