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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-rnegex 7401* Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7361. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Axiomax-precex 7402* Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7362. (Contributed by Jim Kingdon, 6-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
 
Axiomax-cnre 7403* 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 7363. For naming consistency, use cnre 7431 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Axiomax-pre-ltirr 7404 Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7404. (Contributed by Jim Kingdon, 12-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 
Axiomax-pre-ltwlin 7405 Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7365. (Contributed by Jim Kingdon, 12-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
 
Axiomax-pre-lttrn 7406 Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 7366. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Axiomax-pre-apti 7407 Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7367. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
 
Axiomax-pre-ltadd 7408 Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 7368. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
 
Axiomax-pre-mulgt0 7409 The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7369. (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
Axiomax-pre-mulext 7410 Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, justified by theorem axpre-mulext 7370

(Contributed by Jim Kingdon, 18-Feb-2020.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))
 
Axiomax-arch 7411* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 7373.

This axiom should not be used directly; instead use arch 8606 (which is the same, but stated in terms of and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

(𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
 
Axiomax-caucvg 7412* Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 7382.

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 10331 (which is the same, but stated in terms of the and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)

𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))
 
3.2  Derive the basic properties from the field axioms
 
3.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 7413 Alias for ax-cnex 7383. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℂ ∈ V
 
Theoremaddcl 7414 Alias for ax-addcl 7388, for naming consistency with addcli 7439. Use this theorem instead of ax-addcl 7388 or axaddcl 7348. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremreaddcl 7415 Alias for ax-addrcl 7389, for naming consistency with readdcli 7448. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremmulcl 7416 Alias for ax-mulcl 7390, for naming consistency with mulcli 7440. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremremulcl 7417 Alias for ax-mulrcl 7391, for naming consistency with remulcli 7449. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremmulcom 7418 Alias for ax-mulcom 7393, for naming consistency with mulcomi 7441. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddass 7419 Alias for ax-addass 7394, for naming consistency with addassi 7443. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremmulass 7420 Alias for ax-mulass 7395, for naming consistency with mulassi 7444. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremadddi 7421 Alias for ax-distr 7396, for naming consistency with adddii 7445. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremrecn 7422 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
 
Theoremreex 7423 The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V
 
Theoremreelprrecn 7424 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}
 
Theoremcnelprrecn 7425 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}
 
Theoremadddir 7426 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)))
 
Theorem0cn 7427 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
0 ∈ ℂ
 
Theorem0cnd 7428 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℂ)
 
Theoremc0ex 7429 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V
 
Theorem1ex 7430 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V
 
Theoremcnre 7431* Alias for ax-cnre 7403, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremmulid1 7432 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)
 
Theoremmulid2 7433 Identity law for multiplication. Note: see mulid1 7432 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
 
Theorem1re 7434 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
1 ∈ ℝ
 
Theorem0re 7435 0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0 ∈ ℝ
 
Theorem0red 7436 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 0 ∈ ℝ)
 
Theoremmulid1i 7437 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (𝐴 · 1) = 𝐴
 
Theoremmulid2i 7438 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (1 · 𝐴) = 𝐴
 
Theoremaddcli 7439 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) ∈ ℂ
 
Theoremmulcli 7440 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) ∈ ℂ
 
Theoremmulcomi 7441 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) = (𝐵 · 𝐴)
 
Theoremmulcomli 7442 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 · 𝐵) = 𝐶       (𝐵 · 𝐴) = 𝐶
 
Theoremaddassi 7443 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 
Theoremmulassi 7444 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
 
Theoremadddii 7445 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
 
Theoremadddiri 7446 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
 
Theoremrecni 7447 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
𝐴 ∈ ℝ       𝐴 ∈ ℂ
 
Theoremreaddcli 7448 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 + 𝐵) ∈ ℝ
 
Theoremremulcli 7449 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 · 𝐵) ∈ ℝ
 
Theorem1red 7450 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)
 
Theorem1cnd 7451 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)
 
Theoremmulid1d 7452 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)
 
Theoremmulid2d 7453 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1 · 𝐴) = 𝐴)
 
Theoremaddcld 7454 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremmulcld 7455 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremmulcomd 7456 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddassd 7457 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 
Theoremmulassd 7458 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
 
Theoremadddid 7459 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 
Theoremadddird 7460 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
Theoremadddirp1d 7461 Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))
 
Theoremjoinlmuladdmuld 7462 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)
 
Theoremrecnd 7463 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℂ)
 
Theoremreaddcld 7464 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremremulcld 7465 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ)
 
3.2.2  Infinity and the extended real number system
 
Syntaxcpnf 7466 Plus infinity.
class +∞
 
Syntaxcmnf 7467 Minus infinity.
class -∞
 
Syntaxcxr 7468 The set of extended reals (includes plus and minus infinity).
class *
 
Syntaxclt 7469 'Less than' predicate (extended to include the extended reals).
class <
 
Syntaxcle 7470 Extend wff notation to include the 'less than or equal to' relation.
class
 
Definitiondf-pnf 7471 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 7472). We use 𝒫 to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 7476 and mnfnre 7477, and we'll also be able to prove +∞ ≠ -∞.

A simpler possibility is to define +∞ as 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 . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

+∞ = 𝒫
 
Definitiondf-mnf 7472 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in and different from +∞ (see mnfnre 7477). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞
 
Definitiondf-xr 7473 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 ∪ {+∞, -∞})
 
Definitiondf-ltxr 7474* 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, < is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)
< = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
 
Definitiondf-le 7475 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )
 
Theorempnfnre 7476 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ
 
Theoremmnfnre 7477 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ
 
Theoremressxr 7478 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*
 
Theoremrexpssxrxp 7479 The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(ℝ × ℝ) ⊆ (ℝ* × ℝ*)
 
Theoremrexr 7480 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
 
Theorem0xr 7481 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 ∈ ℝ*
 
Theoremrenepnf 7482 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ +∞)
 
Theoremrenemnf 7483 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ -∞)
 
Theoremrexrd 7484 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ*)
 
Theoremrenepnfd 7485 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ +∞)
 
Theoremrenemnfd 7486 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ -∞)
 
Theorempnfxr 7487 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+∞ ∈ ℝ*
 
Theorempnfex 7488 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ ∈ V
 
Theorempnfnemnf 7489 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞
 
Theoremmnfnepnf 7490 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞
 
Theoremmnfxr 7491 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*
 
Theoremrexri 7492 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       𝐴 ∈ ℝ*
 
Theoremrenfdisj 7493 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅
 
Theoremltrelxr 7494 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)
 
Theoremltrel 7495 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <
 
Theoremlerelxr 7496 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)
 
Theoremlerel 7497 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤
 
Theoremxrlenlt 7498 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
 
Theoremltxrlt 7499 The standard less-than < and the extended real less-than < are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
 
3.2.3  Restate the ordering postulates with extended real "less than"
 
Theoremaxltirr 7500 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7404 with ordering on the extended reals. New proofs should use ltnr 7509 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
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