P. 79 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axmulf 7801	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 7798. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7867. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
⊢ · :(ℂ × ℂ)⟶ℂ
Theorem	axaddcom 7802	Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 7844 be used later. Instead, use addcom 8026. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	axmulcom 7803	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7845 be used later. Instead, use mulcom 7873. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	axaddass 7804	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 for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 7846 be used later. Instead, use addass 7874. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	axmulass 7805	Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 7847. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	axdistr 7806	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 7848 be used later. Instead, use adddi 7876. (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 7807	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 7849. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
⊢ ((i · i) + 1) = 0
Theorem	ax0lt1 7808	0 is less than 1. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0lt1 7850. The version of this axiom in the Metamath Proof Explorer reads 1 ≠ 0; here we change it to 0 <ℝ 1. The proof of 0 <ℝ 1 from 1 ≠ 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
⊢ 0 <ℝ 1
Theorem	ax1rid 7809	1 is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7851. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Theorem	ax0id 7810	0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 7852. In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	axrnegex 7811*	Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7853. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	axprecex 7812*	Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 7854. In treatments which assume excluded middle, the 0 <ℝ 𝐴 condition is generally replaced by 𝐴 ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))
Theorem	axcnre 7813*	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, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7855. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	axpre-ltirr 7814	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7856. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
Theorem	axpre-ltwlin 7815	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7857. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
Theorem	axpre-lttrn 7816	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 7858. (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-apti 7817	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 7859. (Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)
Theorem	axpre-ltadd 7818	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 7860. (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 7819	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 7861. (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-mulext 7820	Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7862. (Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Theorem	rereceu 7821*	The reciprocal from axprecex 7812 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Theorem	recriota 7822*	Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
⊢ (𝑁 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
Theorem	axarch 7823*	Archimedean axiom. The Archimedean property is more naturally stated once we have defined ℕ. Unless we find another way to state it, we'll just use the right hand side of dfnn2 8850 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 7863. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
Theorem	peano5nnnn 7824*	Peano's inductive postulate. This is a counterpart to peano5nni 8851 designed for real number axioms which involve natural numbers (notably, axcaucvg 7832). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   ⊢ ((1 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧 + 1) ∈ 𝐴) → 𝑁 ⊆ 𝐴)
Theorem	nnindnn 7825*	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8864 designed for real number axioms which involve natural numbers (notably, axcaucvg 7832). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑧 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑘 ∈ 𝑁 → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ 𝑁 → 𝜏)
Theorem	nntopi 7826*	Mapping from ℕ to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   ⊢ (𝐴 ∈ 𝑁 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)
Theorem	axcaucvglemcl 7827*	Lemma for axcaucvg 7832. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
Theorem	axcaucvglemf 7828*	Lemma for axcaucvg 7832. Mapping to N and R yields a sequence. (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 → 𝐺:N⟶R)
Theorem	axcaucvglemval 7829*	Lemma for axcaucvg 7832. Value of sequence when mapping to 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 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩)
Theorem	axcaucvglemcau 7830*	Lemma for axcaucvg 7832. 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 7831*	Lemma for axcaucvg 7832. 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 7832*	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 7864. (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 7833*	Lemma for axpre-suploc 7834. 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 7741. (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 7834*	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 7865. (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 7835	The complex numbers form a set. Proofs should normally use cnex 7868 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
⊢ ℂ ∈ V
Axiom	ax-resscn 7836	The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 7792. (Contributed by NM, 1-Mar-1995.)
⊢ ℝ ⊆ ℂ
Axiom	ax-1cn 7837	1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 7793. (Contributed by NM, 1-Mar-1995.)
⊢ 1 ∈ ℂ
Axiom	ax-1re 7838	1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 7794. Proofs should use 1re 7889 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
⊢ 1 ∈ ℝ
Axiom	ax-icn 7839	i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 7795. (Contributed by NM, 1-Mar-1995.)
⊢ i ∈ ℂ
Axiom	ax-addcl 7840	Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 7796. Proofs should normally use addcl 7869 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 7841	Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 7797. Proofs should normally use readdcl 7870 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Axiom	ax-mulcl 7842	Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7798. Proofs should normally use mulcl 7871 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Axiom	ax-mulrcl 7843	Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7799. Proofs should normally use remulcl 7872 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Axiom	ax-addcom 7844	Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 7802. Proofs should normally use addcom 8026 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Axiom	ax-mulcom 7845	Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 7803. Proofs should normally use mulcom 7873 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Axiom	ax-addass 7846	Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7804. Proofs should normally use addass 7874 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 7847	Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7805. Proofs should normally use mulass 7875 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 7848	Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 7806. Proofs should normally use adddi 7876 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 7849	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 7807. (Contributed by NM, 29-Jan-1995.)
⊢ ((i · i) + 1) = 0
Axiom	ax-0lt1 7850	0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 7808. Proofs should normally use 0lt1 8016 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
⊢ 0 <ℝ 1
Axiom	ax-1rid 7851	1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 7809. (Contributed by NM, 29-Jan-1995.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Axiom	ax-0id 7852	0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 7810. Proofs should normally use addid1 8027 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Axiom	ax-rnegex 7853*	Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 7811. (Contributed by Eric Schmidt, 21-May-2007.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Axiom	ax-precex 7854*	Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 7812. (Contributed by Jim Kingdon, 6-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))
Axiom	ax-cnre 7855*	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 7813. For naming consistency, use cnre 7886 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 7856	Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 7856. (Contributed by Jim Kingdon, 12-Jan-2020.)
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
Axiom	ax-pre-ltwlin 7857	Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7815. (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 7858	Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 7816. (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 7859	Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7817. (Contributed by Jim Kingdon, 29-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)
Axiom	ax-pre-ltadd 7860	Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 7818. (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Axiom	ax-pre-mulgt0 7861	The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 7819. (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Axiom	ax-pre-mulext 7862	Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, justified by Theorem axpre-mulext 7820 (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 7863*	Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by Theorem axarch 7823. This axiom should not be used directly; instead use arch 9102 (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 7864*	Completeness. Axiom for real and complex numbers, justified by Theorem axcaucvg 7832. 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 10909 (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 7865*	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 7864 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7864. (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 7866	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 7869 should be used. Note that uses of ax-addf 7866 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 7800. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ + :(ℂ × ℂ)⟶ℂ
Axiom	ax-mulf 7867	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 https://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 7842 should be used. Note that uses of ax-mulf 7867 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 7801. (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 7868	Alias for ax-cnex 7835. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 7869	Alias for ax-addcl 7840, for naming consistency with addcli 7894. Use this theorem instead of ax-addcl 7840 or axaddcl 7796. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	readdcl 7870	Alias for ax-addrcl 7841, for naming consistency with readdcli 7903. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	mulcl 7871	Alias for ax-mulcl 7842, for naming consistency with mulcli 7895. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	remulcl 7872	Alias for ax-mulrcl 7843, for naming consistency with remulcli 7904. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	mulcom 7873	Alias for ax-mulcom 7845, for naming consistency with mulcomi 7896. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addass 7874	Alias for ax-addass 7846, for naming consistency with addassi 7898. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulass 7875	Alias for ax-mulass 7847, for naming consistency with mulassi 7899. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddi 7876	Alias for ax-distr 7848, for naming consistency with adddii 7900. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	recn 7877	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 7878	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 7879	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 7880	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	adddir 7881	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 7882	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 7883	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 7884	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	1ex 7885	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	cnre 7886*	Alias for ax-cnre 7855, for naming consistency. (Contributed by NM, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	mulid1 7887	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 7888	Identity law for multiplication. Note: see mulid1 7887 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 7889	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
⊢ 1 ∈ ℝ
Theorem	0re 7890	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 0 ∈ ℝ
Theorem	0red 7891	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	mulid1i 7892	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mulid2i 7893	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 7894	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 7895	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 7896	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 7897	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 7898	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 7899	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 7900	Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))