P. 82 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axcaucvglemcau 8101*	Lemma for axcaucvg 8103. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Theorem	axcaucvglemres 8102*	Lemma for axcaucvg 8103. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
Theorem	axcaucvg 8103*	Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). Because we are stating this axiom before we have introduced notations for ℕ or division, we use 𝑁 for the natural numbers and express a reciprocal in terms of ℩. This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 8135. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
Theorem	axpre-suploclemres 8104*	Lemma for axpre-suploc 8105. The result. The proof just needs to define 𝐵 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ℝ), and apply suplocsr 8012. (Contributed by Jim Kingdon, 24-Jan-2024.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))    &   ⊢ 𝐵 = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
Theorem	axpre-suploc 8105*	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 8136. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
4.1.3  Real and complex number postulates restated as axioms
Axiom	ax-cnex 8106	The complex numbers form a set. Proofs should normally use cnex 8139 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
⊢ ℂ ∈ V
Axiom	ax-resscn 8107	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 8063. (Contributed by NM, 1-Mar-1995.)
⊢ ℝ ⊆ ℂ
Axiom	ax-1cn 8108	1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 8064. (Contributed by NM, 1-Mar-1995.)
⊢ 1 ∈ ℂ
Axiom	ax-1re 8109	1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 8065. Proofs should use 1re 8161 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
⊢ 1 ∈ ℝ
Axiom	ax-icn 8110	i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 8066. (Contributed by NM, 1-Mar-1995.)
⊢ i ∈ ℂ
Axiom	ax-addcl 8111	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 8067. Proofs should normally use addcl 8140 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 8112	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 8068. Proofs should normally use readdcl 8141 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Axiom	ax-mulcl 8113	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 8069. Proofs should normally use mulcl 8142 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Axiom	ax-mulrcl 8114	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 8070. Proofs should normally use remulcl 8143 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Axiom	ax-addcom 8115	Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 8073. Proofs should normally use addcom 8299 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Axiom	ax-mulcom 8116	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8074. Proofs should normally use mulcom 8144 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Axiom	ax-addass 8117	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8075. Proofs should normally use addass 8145 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 8118	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8076. Proofs should normally use mulass 8146 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 8119	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 8077. Proofs should normally use adddi 8147 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 8120	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 8078. (Contributed by NM, 29-Jan-1995.)
⊢ ((i · i) + 1) = 0
Axiom	ax-0lt1 8121	0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 8079. Proofs should normally use 0lt1 8289 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
⊢ 0 <ℝ 1
Axiom	ax-1rid 8122	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 8080. (Contributed by NM, 29-Jan-1995.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Axiom	ax-0id 8123	0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 8081. Proofs should normally use addrid 8300 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Axiom	ax-rnegex 8124*	Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 8082. (Contributed by Eric Schmidt, 21-May-2007.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Axiom	ax-precex 8125*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 8083. (Contributed by Jim Kingdon, 6-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))
Axiom	ax-cnre 8126*	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by Theorem axcnre 8084. For naming consistency, use cnre 8158 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-ltirr 8127	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 8127. (Contributed by Jim Kingdon, 12-Jan-2020.)
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
Axiom	ax-pre-ltwlin 8128	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 8086. (Contributed by Jim Kingdon, 12-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
Axiom	ax-pre-lttrn 8129	Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 8087. (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
Axiom	ax-pre-apti 8130	Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 8088. (Contributed by Jim Kingdon, 29-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)
Axiom	ax-pre-ltadd 8131	Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 8089. (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Axiom	ax-pre-mulgt0 8132	The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 8090. (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Axiom	ax-pre-mulext 8133	Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, justified by Theorem axpre-mulext 8091 (Contributed by Jim Kingdon, 18-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Axiom	ax-arch 8134*	Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by Theorem axarch 8094. This axiom should not be used directly; instead use arch 9382 (which is the same, but stated in terms of ℕ and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
Axiom	ax-caucvg 8135*	Completeness. Axiom for real and complex numbers, justified by Theorem axcaucvg 8103. A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term. This axiom should not be used directly; instead use caucvgre 11513 (which is the same, but stated in terms of the ℕ and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
Axiom	ax-pre-suploc 8136*	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound. Although this and ax-caucvg 8135 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 8135. (Contributed by Jim Kingdon, 23-Jan-2024.)
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
Axiom	ax-addf 8137	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 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 8140 should be used. Note that uses of ax-addf 8137 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 8071. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ + :(ℂ × ℂ)⟶ℂ
Axiom	ax-mulf 8138	Multiplication is an operation on the complex numbers. This axiom tells us that · is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 8364 or eff 12195. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 8142. Note that uses of ax-mulf 8138 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 8152. This axiom is justified by Theorem axmulf 8072. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ · :(ℂ × ℂ)⟶ℂ
4.2  Derive the basic properties from the field axioms
4.2.1  Some deductions from the field axioms for complex numbers
Theorem	cnex 8139	Alias for ax-cnex 8106. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 8140	Alias for ax-addcl 8111, for naming consistency with addcli 8166. Use this theorem instead of ax-addcl 8111 or axaddcl 8067. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	readdcl 8141	Alias for ax-addrcl 8112, for naming consistency with readdcli 8175. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	mulcl 8142	Alias for ax-mulcl 8113, for naming consistency with mulcli 8167. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	remulcl 8143	Alias for ax-mulrcl 8114, for naming consistency with remulcli 8176. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	mulcom 8144	Alias for ax-mulcom 8116, for naming consistency with mulcomi 8168. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addass 8145	Alias for ax-addass 8117, for naming consistency with addassi 8170. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulass 8146	Alias for ax-mulass 8118, for naming consistency with mulassi 8171. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddi 8147	Alias for ax-distr 8119, for naming consistency with adddii 8172. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	recn 8148	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 8149	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 8150	Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℝ ∈ {ℝ, ℂ}
Theorem	cnelprrecn 8151	Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℂ ∈ {ℝ, ℂ}
Theorem	mpomulf 8152*	Multiplication is an operation on complex numbers. Version of ax-mulf 8138 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8113. (Contributed by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	adddir 8153	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 8154	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 8155	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 8156	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	1ex 8157	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	cnre 8158*	Alias for ax-cnre 8126, for naming consistency. (Contributed by NM, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	mulrid 8159	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	mullid 8160	Identity law for multiplication. Note: see mulrid 8159 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 8161	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
⊢ 1 ∈ ℝ
Theorem	0re 8162	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 0 ∈ ℝ
Theorem	0red 8163	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	mulridi 8164	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mullidi 8165	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 8166	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 8167	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 8168	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 8169	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 8170	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 8171	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 8172	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 8173	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 8174	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	readdcli 8175	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℝ
Theorem	remulcli 8176	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℝ
Theorem	1red 8177	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℝ)
Theorem	1cnd 8178	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℂ)
Theorem	mulridd 8179	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐴)
Theorem	mullidd 8180	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	mulid2d 8181	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	addcld 8182	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
Theorem	mulcld 8183	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
Theorem	mulcomd 8184	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addassd 8185	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)))
Theorem	mulassd 8186	Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddid 8187	Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	adddird 8188	Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
Theorem	adddirp1d 8189	Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))
Theorem	joinlmuladdmuld 8190	Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
Theorem	recnd 8191	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	readdcld 8192	Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
Theorem	remulcld 8193	Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ)
4.2.2  Infinity and the extended real number system
Syntax	cpnf 8194	Plus infinity.
class +∞
Syntax	cmnf 8195	Minus infinity.
class -∞
Syntax	cxr 8196	The set of extended reals (includes plus and minus infinity).
class ℝ*
Syntax	clt 8197	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 8198	Extend wff notation to include the 'less than or equal to' relation.
class ≤
Definition	df-pnf 8199	Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in ℝ and different from -∞ (df-mnf 8200). We use 𝒫 ∪ ℂ to make it independent of the construction of ℂ, and Cantor's Theorem will show that it is different from any member of ℂ and therefore ℝ. See pnfnre 8204 and mnfnre 8205, and we'll also be able to prove +∞ ≠ -∞. A simpler possibility is to define +∞ as ℂ and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of ℝ. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
⊢ +∞ = 𝒫 ∪ ℂ
Definition	df-mnf 8200	Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in ℝ and different from +∞ (see mnfnre 8205). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
⊢ -∞ = 𝒫 +∞