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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxpre-lttrn 7101 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7141. (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-apti 7102 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 7142.

(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)

Theoremaxpre-ltadd 7103 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7143. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Theoremaxpre-mulgt0 7104 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 7144. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Theoremaxpre-mulext 7105 Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7145.

(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))

Theoremrereceu 7106* The reciprocal from axprecex 7097 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

Theoremrecriota 7107* Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝑁N → (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Theoremaxarch 7108* Archimedean axiom. The Archimedean property is more naturally stated once we have defined . Unless we find another way to state it, we'll just use the right hand side of dfnn2 8097 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7146. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

(𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)

Theorempeano5nnnn 7109* Peano's inductive postulate. This is a counterpart to peano5nni 8098 designed for real number axioms which involve natural numbers (notably, axcaucvg 7117). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       ((1 ∈ 𝐴 ∧ ∀𝑧𝐴 (𝑧 + 1) ∈ 𝐴) → 𝑁𝐴)

Theoremnnindnn 7110* Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8111 designed for real number axioms which involve natural numbers (notably, axcaucvg 7117). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝑧 = 1 → (𝜑𝜓))    &   (𝑧 = 𝑘 → (𝜑𝜒))    &   (𝑧 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑧 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑘𝑁 → (𝜒𝜃))       (𝐴𝑁𝜏)

Theoremnntopi 7111* Mapping from to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       (𝐴𝑁 → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)

Theoremaxcaucvglemcl 7112* Lemma for axcaucvg 7117. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)       ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)

Theoremaxcaucvglemf 7113* Lemma for axcaucvg 7117. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑𝐺:NR)

Theoremaxcaucvglemval 7114* Lemma for axcaucvg 7117. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩)

Theoremaxcaucvglemcau 7115* Lemma for axcaucvg 7117. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

Theoremaxcaucvglemres 7116* Lemma for axcaucvg 7117. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

Theoremaxcaucvg 7117* 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 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 7147. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

3.1.3  Real and complex number postulates restated as axioms

Axiomax-cnex 7118 The complex numbers form a set. Proofs should normally use cnex 7148 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
ℂ ∈ V

Axiomax-resscn 7119 The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 7079. (Contributed by NM, 1-Mar-1995.)
ℝ ⊆ ℂ

Axiomax-1cn 7120 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 7080. (Contributed by NM, 1-Mar-1995.)
1 ∈ ℂ

Axiomax-1re 7121 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 7081. Proofs should use 1re 7169 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
1 ∈ ℝ

Axiomax-icn 7122 i is a complex number. Axiom for real and complex numbers, justified by theorem axicn 7082. (Contributed by NM, 1-Mar-1995.)
i ∈ ℂ

Axiomax-addcl 7123 Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7083. Proofs should normally use addcl 7149 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 7124 Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 7084. Proofs should normally use readdcl 7150 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Axiomax-mulcl 7125 Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7085. Proofs should normally use mulcl 7151 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Axiomax-mulrcl 7126 Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7086. Proofs should normally use remulcl 7152 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)

Axiomax-addcom 7127 Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7087. Proofs should normally use addcom 7301 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Axiomax-mulcom 7128 Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7088. Proofs should normally use mulcom 7153 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

Axiomax-addass 7129 Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7089. Proofs should normally use addass 7154 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 7130 Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7090. Proofs should normally use mulass 7155 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 7131 Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7091. Proofs should normally use adddi 7156 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 7132 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7092. (Contributed by NM, 29-Jan-1995.)
((i · i) + 1) = 0

Axiomax-0lt1 7133 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7093. Proofs should normally use 0lt1 7292 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
0 < 1

Axiomax-1rid 7134 1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7094. (Contributed by NM, 29-Jan-1995.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)

Axiomax-0id 7135 0 is an identity element for real addition. Axiom for real and complex numbers, justified by theorem ax0id 7095.

Proofs should normally use addid1 7302 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)

(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Axiomax-rnegex 7136* Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7096. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Axiomax-precex 7137* Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7097. (Contributed by Jim Kingdon, 6-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))

Axiomax-cnre 7138* 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 7098. For naming consistency, use cnre 7166 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-ltirr 7139 Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7139. (Contributed by Jim Kingdon, 12-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Axiomax-pre-ltwlin 7140 Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7100. (Contributed by Jim Kingdon, 12-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))

Axiomax-pre-lttrn 7141 Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7101. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Axiomax-pre-apti 7142 Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7102. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)

Axiomax-pre-ltadd 7143 Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 7103. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Axiomax-pre-mulgt0 7144 The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7104. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Axiomax-pre-mulext 7145 Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, justified by theorem axpre-mulext 7105

(Contributed by Jim Kingdon, 18-Feb-2020.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))

Axiomax-arch 7146* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 7108.

This axiom should not be used directly; instead use arch 8341 (which is the same, but stated in terms of and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

(𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)

Axiomax-caucvg 7147* Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 7117.

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 9994 (which is the same, but stated in terms of the and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)

𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))

3.2  Derive the basic properties from the field axioms

3.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 7148 Alias for ax-cnex 7118. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℂ ∈ V

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Theoremmulcl 7151 Alias for ax-mulcl 7125, for naming consistency with mulcli 7175. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Theoremremulcl 7152 Alias for ax-mulrcl 7126, for naming consistency with remulcli 7184. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)

Theoremmulcom 7153 Alias for ax-mulcom 7128, for naming consistency with mulcomi 7176. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulass 7155 Alias for ax-mulass 7130, for naming consistency with mulassi 7179. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddi 7156 Alias for ax-distr 7131, for naming consistency with adddii 7180. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremrecn 7157 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℂ)

Theoremreex 7158 The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V

Theoremreelprrecn 7159 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}

Theoremcnelprrecn 7160 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}

Theoremadddir 7161 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 7162 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
0 ∈ ℂ

Theorem0cnd 7163 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℂ)

Theoremc0ex 7164 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V

Theorem1ex 7165 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V

Theoremcnre 7166* Alias for ax-cnre 7138, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Theoremmulid1 7167 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 7168 Identity law for multiplication. Note: see mulid1 7167 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)

Theorem1re 7169 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
1 ∈ ℝ

Theorem0re 7170 0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0 ∈ ℝ

Theorem0red 7171 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 0 ∈ ℝ)

Theoremmulid1i 7172 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (𝐴 · 1) = 𝐴

Theoremmulid2i 7173 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (1 · 𝐴) = 𝐴

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) ∈ ℂ

Theoremmulcli 7175 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) ∈ ℂ

Theoremmulcomi 7176 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) = (𝐵 · 𝐴)

Theoremmulcomli 7177 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 · 𝐵) = 𝐶       (𝐵 · 𝐴) = 𝐶

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Theoremmulassi 7179 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Theoremadddii 7180 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Theoremadddiri 7181 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))

Theoremrecni 7182 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
𝐴 ∈ ℝ       𝐴 ∈ ℂ

Theoremreaddcli 7183 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 + 𝐵) ∈ ℝ

Theoremremulcli 7184 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 · 𝐵) ∈ ℝ

Theorem1red 7185 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)

Theorem1cnd 7186 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)

Theoremmulid1d 7187 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)

Theoremmulid2d 7188 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1 · 𝐴) = 𝐴)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) ∈ ℂ)

Theoremmulcld 7190 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)

Theoremmulcomd 7191 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulassd 7193 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddid 7194 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremadddird 7195 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremadddirp1d 7196 Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))

Theoremjoinlmuladdmuld 7197 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Theoremrecnd 7198 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℂ)

Theoremreaddcld 7199 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ)

Theoremremulcld 7200 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ)

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