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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | axaddcom 7001 |
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 7041 be used later.
Instead, use addcom 7210.
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 7002 | 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 7042 be used later. Instead, use mulcom 7067. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | axaddass 7003 | 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 7043 be used later. Instead, use addass 7068. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | axmulass 7004 | 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 7044. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | axdistr 7005 | 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 7045 be used later. Instead, use adddi 7070. (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 7006 | 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 7046. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ ((i · i) + 1) = 0 | ||
Theorem | ax0lt1 7007 |
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 7047.
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 7008 | 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 7048. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Theorem | ax0id 7009 |
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 7049.
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 7010* | 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 7050. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Theorem | axprecex 7011* |
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 7051.
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 7012* | 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 7052. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | axpre-ltirr 7013 | 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 7053. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <_{ℝ} 𝐴) | ||
Theorem | axpre-ltwlin 7014 | 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 7054. (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 7015 | 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 7055. (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 7016 |
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 7056.
(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 7017 | 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 7057. (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 7018 | 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 7058. (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 7019 |
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 7059.
(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 7020* | The reciprocal from axprecex 7011 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 <_{ℝ} 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Theorem | recriota 7021* | 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) | ||
Theorem | axarch 7022* |
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 7991 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 7060. (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 7023* | Peano's inductive postulate. This is a counterpart to peano5nni 7992 designed for real number axioms which involve natural numbers (notably, axcaucvg 7031). (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 7024* | Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8005 designed for real number axioms which involve natural numbers (notably, axcaucvg 7031). (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 7025* | 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑧, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ = 𝐴) | ||
Theorem | axcaucvglemcl 7026* | Lemma for axcaucvg 7031. 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐽, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨𝑧, 0_{R}⟩) ∈ R) | ||
Theorem | axcaucvglemf 7027* | Lemma for axcaucvg 7031. 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑗, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨𝑧, 0_{R}⟩)) ⇒ ⊢ (𝜑 → 𝐺:N⟶R) | ||
Theorem | axcaucvglemval 7028* | Lemma for axcaucvg 7031. 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑗, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨𝑧, 0_{R}⟩)) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐽, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐽, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨(𝐺‘𝐽), 0_{R}⟩) | ||
Theorem | axcaucvglemcau 7029* | Lemma for axcaucvg 7031. 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑗, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨𝑧, 0_{R}⟩)) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_{N} 𝑘 → ((𝐺‘𝑛) <_{R} ((𝐺‘𝑘) +_{R} [⟨(⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑛, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑛, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} ) ∧ (𝐺‘𝑘) <_{R} ((𝐺‘𝑛) +_{R} [⟨(⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑛, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑛, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} )))) | ||
Theorem | axcaucvglemres 7030* | Lemma for axcaucvg 7031. 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, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑗, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = ⟨𝑧, 0_{R}⟩)) ⇒ ⊢ (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 7031* |
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 7061. (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))))) | ||
Axiom | ax-cnex 7032 | The complex numbers form a set. Proofs should normally use cnex 7062 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
⊢ ℂ ∈ V | ||
Axiom | ax-resscn 7033 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 6993. (Contributed by NM, 1-Mar-1995.) |
⊢ ℝ ⊆ ℂ | ||
Axiom | ax-1cn 7034 | 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 6994. (Contributed by NM, 1-Mar-1995.) |
⊢ 1 ∈ ℂ | ||
Axiom | ax-1re 7035 | 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 6995. Proofs should use 1re 7083 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
⊢ 1 ∈ ℝ | ||
Axiom | ax-icn 7036 | i is a complex number. Axiom for real and complex numbers, justified by theorem axicn 6996. (Contributed by NM, 1-Mar-1995.) |
⊢ i ∈ ℂ | ||
Axiom | ax-addcl 7037 | Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 6997. Proofs should normally use addcl 7063 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 7038 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 6998. Proofs should normally use readdcl 7064 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Axiom | ax-mulcl 7039 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 6999. Proofs should normally use mulcl 7065 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Axiom | ax-mulrcl 7040 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7000. Proofs should normally use remulcl 7066 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Axiom | ax-addcom 7041 | Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7001. Proofs should normally use addcom 7210 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Axiom | ax-mulcom 7042 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7002. Proofs should normally use mulcom 7067 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Axiom | ax-addass 7043 | Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7003. Proofs should normally use addass 7068 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 7044 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7004. Proofs should normally use mulass 7069 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 7045 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7005. Proofs should normally use adddi 7070 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 7046 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7006. (Contributed by NM, 29-Jan-1995.) |
⊢ ((i · i) + 1) = 0 | ||
Theorem | ax-0lt1 7047 | 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7007. Proofs should normally use 0lt1 7201 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) |
⊢ 0 <_{ℝ} 1 | ||
Axiom | ax-1rid 7048 | 1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7008. (Contributed by NM, 29-Jan-1995.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Axiom | ax-0id 7049 |
0 is an identity element for real addition. Axiom for
real and
complex numbers, justified by theorem ax0id 7009.
Proofs should normally use addid1 7211 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | ||
Axiom | ax-rnegex 7050* | Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7010. (Contributed by Eric Schmidt, 21-May-2007.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Axiom | ax-precex 7051* | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7011. (Contributed by Jim Kingdon, 6-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 <_{ℝ} 𝐴) → ∃𝑥 ∈ ℝ (0 <_{ℝ} 𝑥 ∧ (𝐴 · 𝑥) = 1)) | ||
Axiom | ax-cnre 7052* | 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 7012. For naming consistency, use cnre 7080 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 7053 | Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7053. (Contributed by Jim Kingdon, 12-Jan-2020.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <_{ℝ} 𝐴) | ||
Axiom | ax-pre-ltwlin 7054 | Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7014. (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 7055 | Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7015. (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 7056 | Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7016. (Contributed by Jim Kingdon, 29-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <_{ℝ} 𝐵 ∨ 𝐵 <_{ℝ} 𝐴)) → 𝐴 = 𝐵) | ||
Axiom | ax-pre-ltadd 7057 | Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 7017. (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <_{ℝ} 𝐵 → (𝐶 + 𝐴) <_{ℝ} (𝐶 + 𝐵))) | ||
Axiom | ax-pre-mulgt0 7058 | The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7018. (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <_{ℝ} 𝐴 ∧ 0 <_{ℝ} 𝐵) → 0 <_{ℝ} (𝐴 · 𝐵))) | ||
Axiom | ax-pre-mulext 7059 |
Strong extensionality of multiplication (expressed in terms of <_{ℝ}).
Axiom for real and complex numbers, justified by theorem axpre-mulext 7019
(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 7060* |
Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
real and complex numbers, justified by theorem axarch 7022.
This axiom should not be used directly; instead use arch 8235 (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 7061* |
Completeness. Axiom for real and complex numbers, justified by theorem
axcaucvg 7031.
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 9807 (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))))) | ||
Theorem | cnex 7062 | Alias for ax-cnex 7032. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ℂ ∈ V | ||
Theorem | addcl 7063 | Alias for ax-addcl 7037, for naming consistency with addcli 7088. Use this theorem instead of ax-addcl 7037 or axaddcl 6997. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | readdcl 7064 | Alias for ax-addrcl 7038, for naming consistency with readdcli 7097. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | mulcl 7065 | Alias for ax-mulcl 7039, for naming consistency with mulcli 7089. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | remulcl 7066 | Alias for ax-mulrcl 7040, for naming consistency with remulcli 7098. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Theorem | mulcom 7067 | Alias for ax-mulcom 7042, for naming consistency with mulcomi 7090. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addass 7068 | Alias for ax-addass 7043, for naming consistency with addassi 7092. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | mulass 7069 | Alias for ax-mulass 7044, for naming consistency with mulassi 7093. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | adddi 7070 | Alias for ax-distr 7045, for naming consistency with adddii 7094. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | recn 7071 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | ||
Theorem | reex 7072 | The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ℝ ∈ V | ||
Theorem | reelprrecn 7073 | 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 7074 | 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 7075 | 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 7076 | 0 is a complex number. (Contributed by NM, 19-Feb-2005.) |
⊢ 0 ∈ ℂ | ||
Theorem | 0cnd 7077 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℂ) | ||
Theorem | c0ex 7078 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ∈ V | ||
Theorem | 1ex 7079 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 1 ∈ V | ||
Theorem | cnre 7080* | Alias for ax-cnre 7052, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | mulid1 7081 | 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 7082 | Identity law for multiplication. Note: see mulid1 7081 for commuted version. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | ||
Theorem | 1re 7083 | 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) |
⊢ 1 ∈ ℝ | ||
Theorem | 0re 7084 | 0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ 0 ∈ ℝ | ||
Theorem | 0red 7085 | 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℝ) | ||
Theorem | mulid1i 7086 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 1) = 𝐴 | ||
Theorem | mulid2i 7087 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (1 · 𝐴) = 𝐴 | ||
Theorem | addcli 7088 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℂ | ||
Theorem | mulcli 7089 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℂ | ||
Theorem | mulcomi 7090 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) | ||
Theorem | mulcomli 7091 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 · 𝐵) = 𝐶 ⇒ ⊢ (𝐵 · 𝐴) = 𝐶 | ||
Theorem | addassi 7092 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) | ||
Theorem | mulassi 7093 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) | ||
Theorem | adddii 7094 | 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 7095 | 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 7096 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | readdcli 7097 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℝ | ||
Theorem | remulcli 7098 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℝ | ||
Theorem | 1red 7099 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℝ) | ||
Theorem | 1cnd 7100 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℂ) |
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