P. 101 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfcnqs 10001	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ 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 ℂ 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 )
Theorem	addcnsrec 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.)
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]◡ E + [⟨𝐶, 𝐷⟩]◡ E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩]◡ E )
Theorem	mulcnsrec 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.)
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨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
Theorem	axaddf 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
Theorem	axmulf 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
Theorem	axcnex 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
Theorem	axresscn 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
Theorem	ax1cn 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
Theorem	axicn 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
Theorem	axaddcl 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)
Theorem	axaddrcl 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)
Theorem	axmulcl 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)
Theorem	axmulrcl 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)
Theorem	axmulcom 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))
Theorem	axaddass 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)))
Theorem	axmulass 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)))
Theorem	axdistr 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)))
Theorem	axi2m1 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
Theorem	ax1ne0 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
Theorem	ax1rid 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)
Theorem	axrnegex 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)
Theorem	axrrecex 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)
Theorem	axcnre 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)))
Theorem	axpre-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 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Theorem	axpre-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 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
Theorem	axpre-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 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Theorem	axpre-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 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Theorem	axpre-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 ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
Theorem	wuncn 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.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → ℂ ∈ 𝑈)
5.1.3  Real and complex number postulates restated as axioms
Axiom	ax-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
Axiom	ax-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
Axiom	ax-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
Axiom	ax-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
Axiom	ax-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)
Axiom	ax-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)
Axiom	ax-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)
Axiom	ax-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)
Axiom	ax-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))
Axiom	ax-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)))
Axiom	ax-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)))
Axiom	ax-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)))
Axiom	ax-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
Axiom	ax-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
Axiom	ax-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)
Axiom	ax-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)
Axiom	ax-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)
Axiom	ax-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)))
Axiom	ax-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 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Axiom	ax-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 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
Axiom	ax-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 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Axiom	ax-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 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Axiom	ax-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 ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
Axiom	ax-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
Axiom	ax-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
Theorem	cnex 10055	Alias for ax-cnex 10030. See also cnexALT 11866. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 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)
Theorem	readdcl 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)
Theorem	mulcl 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)
Theorem	remulcl 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)
Theorem	mulcom 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))
Theorem	addass 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)))
Theorem	mulass 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)))
Theorem	adddi 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)))
Theorem	recn 10064	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 10065	The real numbers form a set. See also reexALT 11864. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 10066	Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℝ ∈ {ℝ, ℂ}
Theorem	cnelprrecn 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}
Theorem	elimne0 10068	Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Theorem	adddir 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)))
Theorem	0cn 10070	0 is a complex number. See also 0cnALT 10308. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 10071	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 10072	0 is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	1ex 10073	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	cnre 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)))
Theorem	mulid1 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)
Theorem	mulid2 10076	Identity law for multiplication. Note: see mulid1 10075 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 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
Theorem	0re 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
Theorem	0red 10079	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	mulid1i 10080	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mulid2i 10081	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 10082	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 10083	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 10084	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 10085	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 10086	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 10087	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 10088	Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Theorem	adddiri 10089	Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
Theorem	recni 10090	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	readdcli 10091	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℝ
Theorem	remulcli 10092	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℝ
Theorem	1red 10093	1 is an real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℝ)
Theorem	1cnd 10094	1 is a complex number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℂ)
Theorem	mulid1d 10095	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐴)
Theorem	mulid2d 10096	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	addcld 10097	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
Theorem	mulcld 10098	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
Theorem	mulcomd 10099	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addassd 10100	Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))