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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegexsr 11101* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴 ∈ R β†’ βˆƒπ‘₯ ∈ R (𝐴 +R π‘₯) = 0R)
 
Theoremrecexsrlem 11102* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(0R <R 𝐴 β†’ βˆƒπ‘₯ ∈ R (𝐴 Β·R π‘₯) = 1R)
 
Theoremaddgt0sr 11103 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐡) β†’ 0R <R (𝐴 +R 𝐡))
 
Theoremmulgt0sr 11104 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐡) β†’ 0R <R (𝐴 Β·R 𝐡))
 
Theoremsqgt0sr 11105 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((𝐴 ∈ R ∧ 𝐴 β‰  0R) β†’ 0R <R (𝐴 Β·R 𝐴))
 
Theoremrecexsr 11106* 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.)
((𝐴 ∈ R ∧ 𝐴 β‰  0R) β†’ βˆƒπ‘₯ ∈ R (𝐴 Β·R π‘₯) = 1R)
 
Theoremmappsrpr 11107 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.)
𝐢 ∈ R    β‡’   ((𝐢 +R -1R) <R (𝐢 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴 ∈ P)
 
Theoremltpsrpr 11108 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.)
𝐢 ∈ R    β‡’   ((𝐢 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐢 +R [⟨𝐡, 1P⟩] ~R ) ↔ 𝐴<P 𝐡)
 
Theoremmap2psrpr 11109* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐢 ∈ R    β‡’   ((𝐢 +R -1R) <R 𝐴 ↔ βˆƒπ‘₯ ∈ P (𝐢 +R [⟨π‘₯, 1P⟩] ~R ) = 𝐴)
 
Theoremsupsrlem 11110* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐡 = {𝑀 ∣ (𝐢 +R [βŸ¨π‘€, 1P⟩] ~R ) ∈ 𝐴}    &   πΆ ∈ R    β‡’   ((𝐢 ∈ 𝐴 ∧ 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)))
 
Theoremsupsr 11111* 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.)
((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)))
 
Syntaxcc 11112 Class of complex numbers.
class β„‚
 
Syntaxcr 11113 Class of real numbers.
class ℝ
 
Syntaxcc0 11114 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 11115 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 11116 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 11117 Addition on complex numbers.
class +
 
Syntaxcltrr 11118 'Less than' predicate (defined over real subset of complex numbers).
class <ℝ
 
Syntaxcmul 11119 Multiplication on complex numbers. The token Β· is a center dot.
class Β·
 
Definitiondf-c 11120 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 11147. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
β„‚ = (R Γ— R)
 
Definitiondf-0 11121 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
0 = ⟨0R, 0R⟩
 
Definitiondf-1 11122 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
1 = ⟨1R, 0R⟩
 
Definitiondf-i 11123 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
i = ⟨0R, 1R⟩
 
Definitiondf-r 11124 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℝ = (R Γ— {0R})
 
Definitiondf-add 11125* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
+ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ βˆƒπ‘€βˆƒπ‘£βˆƒπ‘’βˆƒπ‘“((π‘₯ = βŸ¨π‘€, π‘£βŸ© ∧ 𝑦 = βŸ¨π‘’, π‘“βŸ©) ∧ 𝑧 = ⟨(𝑀 +R 𝑒), (𝑣 +R 𝑓)⟩))}
 
Definitiondf-mul 11126* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Β· = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) ∧ βˆƒπ‘€βˆƒπ‘£βˆƒπ‘’βˆƒπ‘“((π‘₯ = βŸ¨π‘€, π‘£βŸ© ∧ 𝑦 = βŸ¨π‘’, π‘“βŸ©) ∧ 𝑧 = ⟨((𝑀 Β·R 𝑒) +R (-1R Β·R (𝑣 Β·R 𝑓))), ((𝑣 Β·R 𝑒) +R (𝑀 Β·R 𝑓))⟩))}
 
Definitiondf-lt 11127* Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 11258. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
<ℝ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ βˆƒπ‘§βˆƒπ‘€((π‘₯ = βŸ¨π‘§, 0R⟩ ∧ 𝑦 = βŸ¨π‘€, 0R⟩) ∧ 𝑧 <R 𝑀))}
 
Theoremopelcn 11128 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(⟨𝐴, 𝐡⟩ ∈ β„‚ ↔ (𝐴 ∈ R ∧ 𝐡 ∈ R))
 
Theoremopelreal 11129 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴 ∈ R)
 
Theoremelreal 11130* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ βˆƒπ‘₯ ∈ R ⟨π‘₯, 0R⟩ = 𝐴)
 
Theoremelreal2 11131 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 11132 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 11133 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
<ℝ βŠ† (ℝ Γ— ℝ)
 
Theoremaddcnsr 11134 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 11135 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 11136 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 11137 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 11138 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 11139 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 11140 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 11141 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in β„‚ from those in R. The trick involves qsid 8781, 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 11120), 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 11142 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 11141 and mulcnsrec 11143. (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 11143 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 8780, which shows that the coset of the converse membership relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 11141.

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 10843. (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 11144 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 11150. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 11193. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
+ :(β„‚ Γ— β„‚)βŸΆβ„‚
 
Theoremaxmulf 11145 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 11152. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 11194. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
Β· :(β„‚ Γ— β„‚)βŸΆβ„‚
 
Theoremaxcnex 11146 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12975), 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 11170 instead of cnexALT 12975. Use cnex 11195 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
β„‚ ∈ V
 
Theoremaxresscn 11147 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 11171. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ βŠ† β„‚
 
Theoremax1cn 11148 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 11172. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ β„‚
 
Theoremaxicn 11149 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 11173. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ β„‚
 
Theoremaxaddcl 11150 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 11174 be used later. Instead, in most cases use addcl 11196. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 + 𝐡) ∈ β„‚)
 
Theoremaxaddrcl 11151 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 11175 be used later. Instead, in most cases use readdcl 11197. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Theoremaxmulcl 11152 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 11176 be used later. Instead, in most cases use mulcl 11198. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Theoremaxmulrcl 11153 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 11177 be used later. Instead, in most cases use remulcl 11199. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Theoremaxmulcom 11154 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 11178 be used later. Instead, use mulcom 11200. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Theoremaxaddass 11155 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 11179 be used later. Instead, use addass 11201. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢)))
 
Theoremaxmulass 11156 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 11180. (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 11157 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 11181 be used later. Instead, use adddi 11203. (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 11158 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 11182. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
((i Β· i) + 1) = 0
 
Theoremax1ne0 11159 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 11183. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
1 β‰  0
 
Theoremax1rid 11160 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 11217, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11184. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· 1) = 𝐴)
 
Theoremaxrnegex 11161* 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 11185. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 + π‘₯) = 0)
 
Theoremaxrrecex 11162* 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 11186. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 Β· π‘₯) = 1)
 
Theoremaxcnre 11163* 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 11187. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯ + (i Β· 𝑦)))
 
Theoremaxpre-lttri 11164 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 11290. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11188. (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 11165 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 11291. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11189. (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 11166 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 11292. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11190. (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 11167 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 11293. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 11191. (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 11168* 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 11294. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11192. (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 11169 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 11170 The complex numbers form a set. This axiom is redundant - see cnexALT 12975- but we provide this axiom because the justification theorem axcnex 11146 does not use ax-rep 5285 even though the redundancy proof does. Proofs should normally use cnex 11195 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
β„‚ ∈ V
 
Axiomax-resscn 11171 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by Theorem axresscn 11147. (Contributed by NM, 1-Mar-1995.)
ℝ βŠ† β„‚
 
Axiomax-1cn 11172 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by Theorem ax1cn 11148. (Contributed by NM, 1-Mar-1995.)
1 ∈ β„‚
 
Axiomax-icn 11173 i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by Theorem axicn 11149. (Contributed by NM, 1-Mar-1995.)
i ∈ β„‚
 
Axiomax-addcl 11174 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11150. Proofs should normally use addcl 11196 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 11175 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl 11151. Proofs should normally use readdcl 11197 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Axiomax-mulcl 11176 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl 11152. Proofs should normally use mulcl 11198 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Axiomax-mulrcl 11177 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl 11153. Proofs should normally use remulcl 11199 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Axiomax-mulcom 11178 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom 11154. Proofs should normally use mulcom 11200 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Axiomax-addass 11179 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass 11155. Proofs should normally use addass 11201 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 11180 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11156. Proofs should normally use mulass 11202 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 11181 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr 11157. Proofs should normally use adddi 11203 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 11182 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 11158. (Contributed by NM, 29-Jan-1995.)
((i Β· i) + 1) = 0
 
Axiomax-1ne0 11183 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by Theorem ax1ne0 11159. (Contributed by NM, 29-Jan-1995.)
1 β‰  0
 
Axiomax-1rid 11184 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by Theorem ax1rid 11160. Weakened from the original axiom in the form of statement in mulrid 11217, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· 1) = 𝐴)
 
Axiomax-rnegex 11185* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11161. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 + π‘₯) = 0)
 
Axiomax-rrecex 11186* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11162. (Contributed by Eric Schmidt, 11-Apr-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ βˆƒπ‘₯ ∈ ℝ (𝐴 Β· π‘₯) = 1)
 
Axiomax-cnre 11187* 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 11163. For naming consistency, use cnre 11216 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 11188 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11164. Note: The more general version for extended reals is axlttri 11290. Normally new proofs would use xrlttri 13123. (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 11189 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11165. Note: The more general version for extended reals is axlttrn 11291. Normally new proofs would use lttr 11295. (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 11190 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11166. Normally new proofs would use axltadd 11292. (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 11191 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11167. Normally new proofs would use axmulgt0 11293. (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 11192* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11168. Note: Normally new proofs would use axsup 11294. (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 11193 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 11196 should be used. Note that uses of ax-addf 11193 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 11144. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

+ :(β„‚ Γ— β„‚)βŸΆβ„‚
 
Axiomax-mulf 11194 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 11176 should be used. Note that uses of ax-mulf 11194 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 11145. (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 11195 Alias for ax-cnex 11170. See also cnexALT 12975. (Contributed by Mario Carneiro, 17-Nov-2014.)
β„‚ ∈ V
 
Theoremaddcl 11196 Alias for ax-addcl 11174, for naming consistency with addcli 11225. Use this theorem instead of ax-addcl 11174 or axaddcl 11150. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 + 𝐡) ∈ β„‚)
 
Theoremreaddcl 11197 Alias for ax-addrcl 11175, for naming consistency with readdcli 11234. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + 𝐡) ∈ ℝ)
 
Theoremmulcl 11198 Alias for ax-mulcl 11176, for naming consistency with mulcli 11226. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) ∈ β„‚)
 
Theoremremulcl 11199 Alias for ax-mulrcl 11177, for naming consistency with remulcli 11235. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β· 𝐡) ∈ ℝ)
 
Theoremmulcom 11200 Alias for ax-mulcom 11178, for naming consistency with mulcomi 11227. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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