HomeHome Metamath Proof Explorer
Theorem List (p. 101 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfcnqs 10001 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 7856, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that is a quotient set, even though it is not (compare df-c 9980), and allows us 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.)
ℂ = ((R × R) / E )
 
Theoremaddcnsrec 10002 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 10001 and mulcnsrec 10003. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )
 
Theoremmulcnsrec 10003 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 7855, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 10001.

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 9704. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = [⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩] E )
 
5.1.2  Final derivation of real and complex number postulates
 
Theoremaxaddf 10004 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 10010. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 10053. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
+ :(ℂ × ℂ)⟶ℂ
 
Theoremaxmulf 10005 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 10012. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10054. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
· :(ℂ × ℂ)⟶ℂ
 
Theoremaxcnex 10006 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 11866), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4804 in later theorems by invoking the axiom ax-cnex 10030 instead of cnexALT 11866. Use cnex 10055 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremaxresscn 10007 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 10031. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ ⊆ ℂ
 
Theoremax1cn 10008 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 10032. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ ℂ
 
Theoremaxicn 10009 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 10033. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ ℂ
 
Theoremaxaddcl 10010 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 10034 be used later. Instead, in most cases use addcl 10056. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremaxaddrcl 10011 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 10035 be used later. Instead, in most cases use readdcl 10057. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremaxmulcl 10012 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 10036 be used later. Instead, in most cases use mulcl 10058. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremaxmulrcl 10013 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 10037 be used later. Instead, in most cases use remulcl 10059. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremaxmulcom 10014 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 10038 be used later. Instead, use mulcom 10060. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaxaddass 10015 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 10039 be used later. Instead, use addass 10061. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremaxmulass 10016 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 10040. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremaxdistr 10017 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 10041 be used later. Instead, use adddi 10063. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremaxi2m1 10018 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 10042. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
((i · i) + 1) = 0
 
Theoremax1ne0 10019 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 10043. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
1 ≠ 0
 
Theoremax1rid 10020 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 10075, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10044. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
 
Theoremaxrnegex 10021* 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 10045. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremaxrrecex 10022* 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 10046. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremaxcnre 10023* 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 10047. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremaxpre-lttri 10024 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 10147. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10048. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
 
Theoremaxpre-lttrn 10025 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 10148. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10049. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremaxpre-ltadd 10026 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 10149. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10050. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
 
Theoremaxpre-mulgt0 10027 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 10150. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10051. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
Theoremaxpre-sup 10028* 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 10151. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10052. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremwuncn 10029 A weak universe containing ω 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.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       (𝜑 → ℂ ∈ 𝑈)
 
5.1.3  Real and complex number postulates restated as axioms
 
Axiomax-cnex 10030 The complex numbers form a set. This axiom is redundant - see cnexALT 11866- but we provide this axiom because the justification theorem axcnex 10006 does not use ax-rep 4804 even though the redundancy proof does. Proofs should normally use cnex 10055 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
ℂ ∈ V
 
Axiomax-resscn 10031 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 10007. (Contributed by NM, 1-Mar-1995.)
ℝ ⊆ ℂ
 
Axiomax-1cn 10032 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 10008. (Contributed by NM, 1-Mar-1995.)
1 ∈ ℂ
 
Axiomax-icn 10033 i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 10009. (Contributed by NM, 1-Mar-1995.)
i ∈ ℂ
 
Axiomax-addcl 10034 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10010. Proofs should normally use addcl 10056 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Axiomax-addrcl 10035 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10011. Proofs should normally use readdcl 10057 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Axiomax-mulcl 10036 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 10012. Proofs should normally use mulcl 10058 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Axiomax-mulrcl 10037 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 10013. Proofs should normally use remulcl 10059 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Axiomax-mulcom 10038 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 10014. Proofs should normally use mulcom 10060 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Axiomax-addass 10039 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 10015. Proofs should normally use addass 10061 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Axiomax-mulass 10040 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10016. Proofs should normally use mulass 10062 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Axiomax-distr 10041 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 10017. Proofs should normally use adddi 10063 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Axiomax-i2m1 10042 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 10018. (Contributed by NM, 29-Jan-1995.)
((i · i) + 1) = 0
 
Axiomax-1ne0 10043 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 10019. (Contributed by NM, 29-Jan-1995.)
1 ≠ 0
 
Axiomax-1rid 10044 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 10020. Weakened from the original axiom in the form of statement in mulid1 10075, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
 
Axiomax-rnegex 10045* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 10021. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Axiomax-rrecex 10046* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 10022. (Contributed by Eric Schmidt, 11-Apr-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Axiomax-cnre 10047* 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 10023. For naming consistency, use cnre 10074 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Axiomax-pre-lttri 10048 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 10024. Note: The more general version for extended reals is axlttri 10147. Normally new proofs would use xrlttri 12010. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
 
Axiomax-pre-lttrn 10049 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 10025. Note: The more general version for extended reals is axlttrn 10148. Normally new proofs would use lttr 10152. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Axiomax-pre-ltadd 10050 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 10026. Normally new proofs would use axltadd 10149. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
 
Axiomax-pre-mulgt0 10051 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 10027. Normally new proofs would use axmulgt0 10150. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
Axiomax-pre-sup 10052* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 10028. Note: Normally new proofs would use axsup 10151. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Axiomax-addf 10053 Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://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 10056 should be used. Note that uses of ax-addf 10053 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 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 10004. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

+ :(ℂ × ℂ)⟶ℂ
 
Axiomax-mulf 10054 Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://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 10036 should be used. Note that uses of ax-mulf 10054 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 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 axmulf 10005. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

· :(ℂ × ℂ)⟶ℂ
 
5.2  Derive the basic properties from the field axioms
 
5.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 10055 Alias for ax-cnex 10030. See also cnexALT 11866. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℂ ∈ V
 
Theoremaddcl 10056 Alias for ax-addcl 10034, for naming consistency with addcli 10082. Use this theorem instead of ax-addcl 10034 or axaddcl 10010. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremreaddcl 10057 Alias for ax-addrcl 10035, for naming consistency with readdcli 10091. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremmulcl 10058 Alias for ax-mulcl 10036, for naming consistency with mulcli 10083. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremremulcl 10059 Alias for ax-mulrcl 10037, for naming consistency with remulcli 10092. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremmulcom 10060 Alias for ax-mulcom 10038, for naming consistency with mulcomi 10084. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddass 10061 Alias for ax-addass 10039, for naming consistency with addassi 10086. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremmulass 10062 Alias for ax-mulass 10040, for naming consistency with mulassi 10087. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremadddi 10063 Alias for ax-distr 10041, for naming consistency with adddii 10088. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremrecn 10064 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
 
Theoremreex 10065 The real numbers form a set. See also reexALT 11864. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V
 
Theoremreelprrecn 10066 Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.)
ℝ ∈ {ℝ, ℂ}
 
Theoremcnelprrecn 10067 Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.)
ℂ ∈ {ℝ, ℂ}
 
Theoremelimne0 10068 Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
 
Theoremadddir 10069 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
Theorem0cn 10070 0 is a complex number. See also 0cnALT 10308. (Contributed by NM, 19-Feb-2005.)
0 ∈ ℂ
 
Theorem0cnd 10071 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℂ)
 
Theoremc0ex 10072 0 is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V
 
Theorem1ex 10073 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V
 
Theoremcnre 10074* Alias for ax-cnre 10047, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremmulid1 10075 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
 
Theoremmulid2 10076 Identity law for multiplication. Note: see mulid1 10075 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
 
Theorem1re 10077 1 is a real number. 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 10032, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
1 ∈ ℝ
 
Theorem0re 10078 0 is a real number. See also 0reALT 10416. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0 ∈ ℝ
 
Theorem0red 10079 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 0 ∈ ℝ)
 
Theoremmulid1i 10080 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (𝐴 · 1) = 𝐴
 
Theoremmulid2i 10081 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (1 · 𝐴) = 𝐴
 
Theoremaddcli 10082 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) ∈ ℂ
 
Theoremmulcli 10083 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) ∈ ℂ
 
Theoremmulcomi 10084 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) = (𝐵 · 𝐴)
 
Theoremmulcomli 10085 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 · 𝐵) = 𝐶       (𝐵 · 𝐴) = 𝐶
 
Theoremaddassi 10086 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 
Theoremmulassi 10087 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
 
Theoremadddii 10088 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
 
Theoremadddiri 10089 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
 
Theoremrecni 10090 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
𝐴 ∈ ℝ       𝐴 ∈ ℂ
 
Theoremreaddcli 10091 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 + 𝐵) ∈ ℝ
 
Theoremremulcli 10092 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 · 𝐵) ∈ ℝ
 
Theorem1red 10093 1 is an real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)
 
Theorem1cnd 10094 1 is a complex number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)
 
Theoremmulid1d 10095 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)
 
Theoremmulid2d 10096 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1 · 𝐴) = 𝐴)
 
Theoremaddcld 10097 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremmulcld 10098 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremmulcomd 10099 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddassd 10100 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >