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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))) | ||
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 | ||
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)) |
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