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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-mulcom 10601 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 10577. Proofs should normally use mulcom 10623 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Axiom | ax-addass 10602 | Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 10578. Proofs should normally use addass 10624 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 10603 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10579. Proofs should normally use mulass 10625 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 10604 | Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 10580. Proofs should normally use adddi 10626 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 10605 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 10581. (Contributed by NM, 29-Jan-1995.) |
⊢ ((i · i) + 1) = 0 | ||
Axiom | ax-1ne0 10606 | 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 10582. (Contributed by NM, 29-Jan-1995.) |
⊢ 1 ≠ 0 | ||
Axiom | ax-1rid 10607 | 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 10583. Weakened from the original axiom in the form of statement in mulid1 10639, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | ||
Axiom | ax-rnegex 10608* | Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 10584. (Contributed by Eric Schmidt, 21-May-2007.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | ||
Axiom | ax-rrecex 10609* | Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 10585. (Contributed by Eric Schmidt, 11-Apr-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Axiom | ax-cnre 10610* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 10586. For naming consistency, use cnre 10638 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-lttri 10611 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 10587. Note: The more general version for extended reals is axlttri 10712. Normally new proofs would use xrlttri 12533. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) | ||
Axiom | ax-pre-lttrn 10612 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 10588. Note: The more general version for extended reals is axlttrn 10713. Normally new proofs would use lttr 10717. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) | ||
Axiom | ax-pre-ltadd 10613 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 10589. Normally new proofs would use axltadd 10714. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) | ||
Axiom | ax-pre-mulgt0 10614 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 10590. Normally new proofs would use axmulgt0 10715. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) | ||
Axiom | ax-pre-sup 10615* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 10591. Note: Normally new proofs would use axsup 10716. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))) | ||
Axiom | ax-addf 10616 |
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 10619 should be used. Note that uses of ax-addf 10616 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 10567. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
⊢ + :(ℂ × ℂ)⟶ℂ | ||
Axiom | ax-mulf 10617 |
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 10599 should be used. Note that uses of ax-mulf 10617
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 10568. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.) |
⊢ · :(ℂ × ℂ)⟶ℂ | ||
Theorem | cnex 10618 | Alias for ax-cnex 10593. See also cnexALT 12386. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ℂ ∈ V | ||
Theorem | addcl 10619 | Alias for ax-addcl 10597, for naming consistency with addcli 10647. Use this theorem instead of ax-addcl 10597 or axaddcl 10573. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | readdcl 10620 | Alias for ax-addrcl 10598, for naming consistency with readdcli 10656. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | mulcl 10621 | Alias for ax-mulcl 10599, for naming consistency with mulcli 10648. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | remulcl 10622 | Alias for ax-mulrcl 10600, for naming consistency with remulcli 10657. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | ||
Theorem | mulcom 10623 | Alias for ax-mulcom 10601, for naming consistency with mulcomi 10649. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addass 10624 | Alias for ax-addass 10602, for naming consistency with addassi 10651. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | mulass 10625 | Alias for ax-mulass 10603, for naming consistency with mulassi 10652. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | adddi 10626 | Alias for ax-distr 10604, for naming consistency with adddii 10653. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | recn 10627 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | ||
Theorem | reex 10628 | The real numbers form a set. See also reexALT 12384. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ℝ ∈ V | ||
Theorem | reelprrecn 10629 | Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ ℝ ∈ {ℝ, ℂ} | ||
Theorem | cnelprrecn 10630 | Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ ℂ ∈ {ℝ, ℂ} | ||
Theorem | elimne0 10631 | Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.) |
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 | ||
Theorem | adddir 10632 | 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 10633 | Zero is a complex number. See also 0cnALT 10874. (Contributed by NM, 19-Feb-2005.) |
⊢ 0 ∈ ℂ | ||
Theorem | 0cnd 10634 | Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℂ) | ||
Theorem | c0ex 10635 | Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ∈ V | ||
Theorem | 1cnd 10636 | One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℂ) | ||
Theorem | 1ex 10637 | One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 1 ∈ V | ||
Theorem | cnre 10638* | Alias for ax-cnre 10610, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | mulid1 10639 | The number 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 10640 | Identity law for multiplication. See mulid1 10639 for commuted version. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | ||
Theorem | 1re 10641 | The number 1 is real. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 10595, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ 1 ∈ ℝ | ||
Theorem | 1red 10642 | The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℝ) | ||
Theorem | 0re 10643 | The number 0 is real. Remark: the first step could also be ax-icn 10596. See also 0reALT 10983. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 11-Oct-2022.) |
⊢ 0 ∈ ℝ | ||
Theorem | 0red 10644 | The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℝ) | ||
Theorem | mulid1i 10645 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 1) = 𝐴 | ||
Theorem | mulid2i 10646 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (1 · 𝐴) = 𝐴 | ||
Theorem | addcli 10647 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℂ | ||
Theorem | mulcli 10648 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℂ | ||
Theorem | mulcomi 10649 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) | ||
Theorem | mulcomli 10650 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 · 𝐵) = 𝐶 ⇒ ⊢ (𝐵 · 𝐴) = 𝐶 | ||
Theorem | addassi 10651 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) | ||
Theorem | mulassi 10652 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) | ||
Theorem | adddii 10653 | 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 10654 | 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 10655 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | readdcli 10656 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℝ | ||
Theorem | remulcli 10657 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℝ | ||
Theorem | mulid1d 10658 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐴) | ||
Theorem | mulid2d 10659 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐴) | ||
Theorem | addcld 10660 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | mulcld 10661 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | mulcomd 10662 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addassd 10663 | Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | mulassd 10664 | Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | adddid 10665 | Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | adddird 10666 | Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | ||
Theorem | adddirp1d 10667 | Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) | ||
Theorem | joinlmuladdmuld 10668 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) | ||
Theorem | recnd 10669 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | readdcld 10670 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | remulcld 10671 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) | ||
Syntax | cpnf 10672 | Plus infinity. |
class +∞ | ||
Syntax | cmnf 10673 | Minus infinity. |
class -∞ | ||
Syntax | cxr 10674 | The set of extended reals (includes plus and minus infinity). |
class ℝ* | ||
Syntax | clt 10675 | 'Less than' predicate (extended to include the extended reals). |
class < | ||
Syntax | cle 10676 | Extend wff notation to include the 'less than or equal to' relation. |
class ≤ | ||
Definition | df-pnf 10677 |
Define plus infinity. Note that the definition is arbitrary, requiring
only that +∞ be a set not in ℝ and different from -∞
(df-mnf 10678). We use 𝒫 ∪ ℂ to make it independent of the
construction of ℂ, and Cantor's Theorem will
show that it is
different from any member of ℂ and therefore
ℝ. See pnfnre 10682,
mnfnre 10684, and pnfnemnf 10696.
A simpler possibility is to define +∞ as ℂ and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of ℝ. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.) |
⊢ +∞ = 𝒫 ∪ ℂ | ||
Definition | df-mnf 10678 | Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in ℝ and different from +∞ (see mnfnre 10684 and pnfnemnf 10696). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.) |
⊢ -∞ = 𝒫 +∞ | ||
Definition | df-xr 10679 | Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.) |
⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | ||
Definition | df-ltxr 10680* | Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, <ℝ is primitive and not necessarily a relation on ℝ. (Contributed by NM, 13-Oct-2005.) |
⊢ < = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) | ||
Definition | df-le 10681 | Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 10727 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.) |
⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | ||
Theorem | pnfnre 10682 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ +∞ ∉ ℝ | ||
Theorem | pnfnre2 10683 | Plus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ ¬ +∞ ∈ ℝ | ||
Theorem | mnfnre 10684 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ -∞ ∉ ℝ | ||
Theorem | ressxr 10685 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ ℝ ⊆ ℝ* | ||
Theorem | rexpssxrxp 10686 | The Cartesian product of standard reals are a subset of the Cartesian product of extended reals. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | ||
Theorem | rexr 10687 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | ||
Theorem | 0xr 10688 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
⊢ 0 ∈ ℝ* | ||
Theorem | renepnf 10689 | No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | ||
Theorem | renemnf 10690 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | ||
Theorem | rexrd 10691 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
Theorem | renepnfd 10692 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | renemnfd 10693 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ -∞) | ||
Theorem | pnfex 10694 | Plus infinity exists. (Contributed by David A. Wheeler, 8-Dec-2018.) (Revised by Steven Nguyen, 7-Dec-2022.) |
⊢ +∞ ∈ V | ||
Theorem | pnfxr 10695 | Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
⊢ +∞ ∈ ℝ* | ||
Theorem | pnfnemnf 10696 | Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.) |
⊢ +∞ ≠ -∞ | ||
Theorem | mnfnepnf 10697 | Minus and plus infinity are different. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -∞ ≠ +∞ | ||
Theorem | mnfxr 10698 | Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ -∞ ∈ ℝ* | ||
Theorem | rexri 10699 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℝ* | ||
Theorem | 1xr 10700 | 1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 1 ∈ ℝ* |
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