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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-0 7301 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
⊢ 0 = ⟨0_{R}, 0_{R}⟩ | ||
Definition | df-1 7302 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
⊢ 1 = ⟨1_{R}, 0_{R}⟩ | ||
Definition | df-i 7303 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) |
⊢ i = ⟨0_{R}, 1_{R}⟩ | ||
Definition | df-r 7304 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ ℝ = (R × {0_{R}}) | ||
Definition | df-add 7305* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) |
⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +_{R} 𝑢), (𝑣 +_{R} 𝑓)⟩))} | ||
Definition | df-mul 7306* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) |
⊢ · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·_{R} 𝑢) +_{R} (-1_{R} ·_{R} (𝑣 ·_{R} 𝑓))), ((𝑣 ·_{R} 𝑢) +_{R} (𝑤 ·_{R} 𝑓))⟩))} | ||
Definition | df-lt 7307* | Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ <_{ℝ} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0_{R}⟩ ∧ 𝑦 = ⟨𝑤, 0_{R}⟩) ∧ 𝑧 <_{R} 𝑤))} | ||
Theorem | opelcn 7308 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | ||
Theorem | opelreal 7309 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ (⟨𝐴, 0_{R}⟩ ∈ ℝ ↔ 𝐴 ∈ R) | ||
Theorem | elreal 7310* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0_{R}⟩ = 𝐴) | ||
Theorem | elrealeu 7311* | The real number mapping in elreal 7310 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R ⟨𝑥, 0_{R}⟩ = 𝐴) | ||
Theorem | elreal2 7312 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
⊢ (𝐴 ∈ ℝ ↔ ((1^{st} ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^{st} ‘𝐴), 0_{R}⟩)) | ||
Theorem | 0ncn 7313 | The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) |
⊢ ¬ ∅ ∈ ℂ | ||
Theorem | ltrelre 7314 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ <_{ℝ} ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 7315 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +_{R} 𝐶), (𝐵 +_{R} 𝐷)⟩) | ||
Theorem | mulcnsr 7316 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·_{R} 𝐶) +_{R} (-1_{R} ·_{R} (𝐵 ·_{R} 𝐷))), ((𝐵 ·_{R} 𝐶) +_{R} (𝐴 ·_{R} 𝐷))⟩) | ||
Theorem | eqresr 7317 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (⟨𝐴, 0_{R}⟩ = ⟨𝐵, 0_{R}⟩ ↔ 𝐴 = 𝐵) | ||
Theorem | addresr 7318 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0_{R}⟩ + ⟨𝐵, 0_{R}⟩) = ⟨(𝐴 +_{R} 𝐵), 0_{R}⟩) | ||
Theorem | mulresr 7319 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0_{R}⟩ · ⟨𝐵, 0_{R}⟩) = ⟨(𝐴 ·_{R} 𝐵), 0_{R}⟩) | ||
Theorem | ltresr 7320 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
⊢ (⟨𝐴, 0_{R}⟩ <_{ℝ} ⟨𝐵, 0_{R}⟩ ↔ 𝐴 <_{R} 𝐵) | ||
Theorem | ltresr2 7321 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_{ℝ} 𝐵 ↔ (1^{st} ‘𝐴) <_{R} (1^{st} ‘𝐵))) | ||
Theorem | dfcnqs 7322 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 6309, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that ℂ is a quotient set, even though it is not (compare df-c 7300), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) |
⊢ ℂ = ((R × R) / ^{◡} E ) | ||
Theorem | addcnsrec 7323 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7322 and mulcnsrec 7324. (Contributed by NM, 13-Aug-1995.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]^{◡} E + [⟨𝐶, 𝐷⟩]^{◡} E ) = [⟨(𝐴 +_{R} 𝐶), (𝐵 +_{R} 𝐷)⟩]^{◡} E ) | ||
Theorem | mulcnsrec 7324 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6308, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7322. (Contributed by NM, 13-Aug-1995.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([⟨𝐴, 𝐵⟩]^{◡} E · [⟨𝐶, 𝐷⟩]^{◡} E ) = [⟨((𝐴 ·_{R} 𝐶) +_{R} (-1_{R} ·_{R} (𝐵 ·_{R} 𝐷))), ((𝐵 ·_{R} 𝐶) +_{R} (𝐴 ·_{R} 𝐷))⟩]^{◡} E ) | ||
Theorem | addvalex 7325 | Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7411. (Contributed by Jim Kingdon, 14-Jul-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V) | ||
Theorem | pitonnlem1 7326* | Lemma for pitonn 7329. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.) |
⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨1_{𝑜}, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨1_{𝑜}, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ = 1 | ||
Theorem | pitonnlem1p1 7327 | Lemma for pitonn 7329. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ P → [⟨(𝐴 +_{P} (1_{P} +_{P} 1_{P})), (1_{P} +_{P} 1_{P})⟩] ~_{R} = [⟨(𝐴 +_{P} 1_{P}), 1_{P}⟩] ~_{R} ) | ||
Theorem | pitonnlem2 7328* | Lemma for pitonn 7329. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.) |
⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐾, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐾, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨(𝐾 +_{N} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨(𝐾 +_{N} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) | ||
Theorem | pitonn 7329* | Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.) |
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑁, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) | ||
Theorem | pitoregt0 7330* | Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → 0 <_{ℝ} ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑁, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) | ||
Theorem | pitore 7331* | Embedding from N to ℝ. Similar to pitonn 7329 but separate in the sense that we have not proved nnssre 8361 yet. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑁, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ ∈ ℝ) | ||
Theorem | recnnre 7332* | Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ ∈ ℝ) | ||
Theorem | peano1nnnn 7333* | One is an element of ℕ. This is a counterpart to 1nn 8368 designed for real number axioms which involve natural numbers (notably, axcaucvg 7379). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ 1 ∈ 𝑁 | ||
Theorem | peano2nnnn 7334* | A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8369 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7379). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁) | ||
Theorem | ltrennb 7335* | Ordering of natural numbers with <_{N} or <_{ℝ}. (Contributed by Jim Kingdon, 13-Jul-2021.) |
⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_{N} 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐽, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐽, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ <_{ℝ} ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐾, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐾, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩)) | ||
Theorem | ltrenn 7336* | Ordering of natural numbers with <_{N} or <_{ℝ}. (Contributed by Jim Kingdon, 12-Jul-2021.) |
⊢ (𝐽 <_{N} 𝐾 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐽, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐽, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ <_{ℝ} ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐾, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝐾, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) | ||
Theorem | recidpipr 7337* | Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.) |
⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑁, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ ·_{P} ⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩) = 1_{P}) | ||
Theorem | recidpirqlemcalc 7338 | Lemma for recidpirq 7339. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.) |
⊢ (𝜑 → 𝐴 ∈ P) & ⊢ (𝜑 → 𝐵 ∈ P) & ⊢ (𝜑 → (𝐴 ·_{P} 𝐵) = 1_{P}) ⇒ ⊢ (𝜑 → ((((𝐴 +_{P} 1_{P}) ·_{P} (𝐵 +_{P} 1_{P})) +_{P} (1_{P} ·_{P} 1_{P})) +_{P} 1_{P}) = ((((𝐴 +_{P} 1_{P}) ·_{P} 1_{P}) +_{P} (1_{P} ·_{P} (𝐵 +_{P} 1_{P}))) +_{P} (1_{P} +_{P} 1_{P}))) | ||
Theorem | recidpirq 7339* | A real number times its reciprocal is one, where reciprocal is expressed with *_{Q}. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑁, 1_{𝑜}⟩] ~_{Q} }, {𝑢 ∣ [⟨𝑁, 1_{𝑜}⟩] ~_{Q} <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} )}, {𝑢 ∣ (*_{Q}‘[⟨𝑁, 1_{𝑜}⟩] ~_{Q} ) <_{Q} 𝑢}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) = 1) | ||
Theorem | axcnex 7340 | The complex numbers form a set. Use cnex 7410 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
⊢ ℂ ∈ V | ||
Theorem | axresscn 7341 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 7381. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
⊢ ℝ ⊆ ℂ | ||
Theorem | ax1cn 7342 | 1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 7382. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
⊢ 1 ∈ ℂ | ||
Theorem | ax1re 7343 |
1 is a real number. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-1re 7383.
In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7382 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
⊢ 1 ∈ ℝ | ||
Theorem | axicn 7344 | i is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 7384. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
⊢ i ∈ ℂ | ||
Theorem | axaddcl 7345 | Closure law for addition of complex numbers. 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-addcl 7385 be used later. Instead, in most cases use addcl 7411. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | axaddrcl 7346 | Closure law for addition in the real subfield of complex numbers. 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-addrcl 7386 be used later. Instead, in most cases use readdcl 7412. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | axmulcl 7347 | Closure law for multiplication of complex numbers. 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-mulcl 7387 be used later. Instead, in most cases use mulcl 7413. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | axmulrcl 7348 | Closure law for multiplication in the real subfield of complex numbers. 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-mulrcl 7388 be used later. Instead, in most cases use remulcl 7414. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Theorem | axaddcom 7349 |
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 7389 be used later.
Instead, use addcom 7563.
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 7350 | 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 7390 be used later. Instead, use mulcom 7415. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | axaddass 7351 | 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 7391 be used later. Instead, use addass 7416. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | axmulass 7352 | 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 7392. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | axdistr 7353 | 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 7393 be used later. Instead, use adddi 7418. (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 7354 | 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 7394. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ ((i · i) + 1) = 0 | ||
Theorem | ax0lt1 7355 |
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 7395.
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 7356 | 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 7396. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Theorem | ax0id 7357 |
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 7397.
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 7358* | 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 7398. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Theorem | axprecex 7359* |
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 7399.
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 7360* | 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 7400. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | axpre-ltirr 7361 | 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 7401. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 <_{ℝ} 𝐴) | ||
Theorem | axpre-ltwlin 7362 | 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 7402. (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 7363 | 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 7403. (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 7364 |
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 7404.
(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 7365 | 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 7405. (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 7366 | 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 7406. (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 7367 |
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 7407.
(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 7368* | The reciprocal from axprecex 7359 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 <_{ℝ} 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Theorem | recriota 7369* | 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 7370* |
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 8359 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 7408. (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 7371* | Peano's inductive postulate. This is a counterpart to peano5nni 8360 designed for real number axioms which involve natural numbers (notably, axcaucvg 7379). (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 7372* | Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8373 designed for real number axioms which involve natural numbers (notably, axcaucvg 7379). (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 7373* | 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 7374* | Lemma for axcaucvg 7379. 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 7375* | Lemma for axcaucvg 7379. 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 7376* | Lemma for axcaucvg 7379. 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 7377* | Lemma for axcaucvg 7379. 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 7378* | Lemma for axcaucvg 7379. 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 7379* |
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 7409. (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 7380 | The complex numbers form a set. Proofs should normally use cnex 7410 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
⊢ ℂ ∈ V | ||
Axiom | ax-resscn 7381 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 7341. (Contributed by NM, 1-Mar-1995.) |
⊢ ℝ ⊆ ℂ | ||
Axiom | ax-1cn 7382 | 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 7342. (Contributed by NM, 1-Mar-1995.) |
⊢ 1 ∈ ℂ | ||
Axiom | ax-1re 7383 | 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 7343. Proofs should use 1re 7431 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
⊢ 1 ∈ ℝ | ||
Axiom | ax-icn 7384 | i is a complex number. Axiom for real and complex numbers, justified by theorem axicn 7344. (Contributed by NM, 1-Mar-1995.) |
⊢ i ∈ ℂ | ||
Axiom | ax-addcl 7385 | Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7345. Proofs should normally use addcl 7411 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 7386 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 7346. Proofs should normally use readdcl 7412 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Axiom | ax-mulcl 7387 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7347. Proofs should normally use mulcl 7413 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Axiom | ax-mulrcl 7388 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7348. Proofs should normally use remulcl 7414 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Axiom | ax-addcom 7389 | Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7349. Proofs should normally use addcom 7563 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Axiom | ax-mulcom 7390 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7350. Proofs should normally use mulcom 7415 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Axiom | ax-addass 7391 | Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7351. Proofs should normally use addass 7416 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 7392 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7352. Proofs should normally use mulass 7417 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 7393 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7353. Proofs should normally use adddi 7418 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 7394 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7354. (Contributed by NM, 29-Jan-1995.) |
⊢ ((i · i) + 1) = 0 | ||
Axiom | ax-0lt1 7395 | 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7355. Proofs should normally use 0lt1 7554 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) |
⊢ 0 <_{ℝ} 1 | ||
Axiom | ax-1rid 7396 | 1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7356. (Contributed by NM, 29-Jan-1995.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Axiom | ax-0id 7397 |
0 is an identity element for real addition. Axiom for
real and
complex numbers, justified by theorem ax0id 7357.
Proofs should normally use addid1 7564 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | ||
Axiom | ax-rnegex 7398* | Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7358. (Contributed by Eric Schmidt, 21-May-2007.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Axiom | ax-precex 7399* | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7359. (Contributed by Jim Kingdon, 6-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 <_{ℝ} 𝐴) → ∃𝑥 ∈ ℝ (0 <_{ℝ} 𝑥 ∧ (𝐴 · 𝑥) = 1)) | ||
Axiom | ax-cnre 7400* | 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 7360. For naming consistency, use cnre 7428 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
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