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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-pre-mulext 6801 Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, justified by theorem axpre-mulext 6772

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

((A B 𝐶 ℝ) → ((A · 𝐶) < (B · 𝐶) → (A < B B < A)))

Axiomax-arch 6802* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 6773.

This axiom should not be used directly; instead use arch 7954 (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 y x (y + 1) x)}A < 𝑛)

3.2  Derive the basic properties from the field axioms

3.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 6803 Alias for ax-cnex 6774. (Contributed by Mario Carneiro, 17-Nov-2014.)
V

((A B ℂ) → (A + B) ℂ)

((A B ℝ) → (A + B) ℝ)

Theoremmulcl 6806 Alias for ax-mulcl 6781, for naming consistency with mulcli 6830. (Contributed by NM, 10-Mar-2008.)
((A B ℂ) → (A · B) ℂ)

Theoremremulcl 6807 Alias for ax-mulrcl 6782, for naming consistency with remulcli 6839. (Contributed by NM, 10-Mar-2008.)
((A B ℝ) → (A · B) ℝ)

Theoremmulcom 6808 Alias for ax-mulcom 6784, for naming consistency with mulcomi 6831. (Contributed by NM, 10-Mar-2008.)
((A B ℂ) → (A · B) = (B · A))

((A B 𝐶 ℂ) → ((A + B) + 𝐶) = (A + (B + 𝐶)))

Theoremmulass 6810 Alias for ax-mulass 6786, for naming consistency with mulassi 6834. (Contributed by NM, 10-Mar-2008.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = (A · (B · 𝐶)))

Theoremadddi 6811 Alias for ax-distr 6787, for naming consistency with adddii 6835. (Contributed by NM, 10-Mar-2008.)
((A B 𝐶 ℂ) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))

Theoremrecn 6812 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(A ℝ → A ℂ)

Theoremreex 6813 The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
V

Theoremreelprrecn 6814 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
{ℝ, ℂ}

Theoremcnelprrecn 6815 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
{ℝ, ℂ}

Theoremadddir 6816 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
((A B 𝐶 ℂ) → ((A + B) · 𝐶) = ((A · 𝐶) + (B · 𝐶)))

Theorem0cn 6817 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
0

Theorem0cnd 6818 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(φ → 0 ℂ)

Theoremc0ex 6819 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
0 V

Theorem1ex 6820 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 V

Theoremcnre 6821* Alias for ax-cnre 6794, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(A ℂ → x y A = (x + (i · y)))

Theoremmulid1 6822 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 6823 Identity law for multiplication. Note: see mulid1 6822 for commuted version. (Contributed by NM, 8-Oct-1999.)
(A ℂ → (1 · A) = A)

Theorem1re 6824 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
1

Theorem0re 6825 0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0

Theorem0red 6826 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(φ → 0 ℝ)

Theoremmulid1i 6827 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
A        (A · 1) = A

Theoremmulid2i 6828 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
A        (1 · A) = A

A     &   B        (A + B)

Theoremmulcli 6830 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
A     &   B        (A · B)

Theoremmulcomi 6831 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
A     &   B        (A · B) = (B · A)

Theoremmulcomli 6832 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
A     &   B     &   (A · B) = 𝐶       (B · A) = 𝐶

A     &   B     &   𝐶        ((A + B) + 𝐶) = (A + (B + 𝐶))

Theoremmulassi 6834 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
A     &   B     &   𝐶        ((A · B) · 𝐶) = (A · (B · 𝐶))

Theoremadddii 6835 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
A     &   B     &   𝐶        (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))

Theoremadddiri 6836 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
A     &   B     &   𝐶        ((A + B) · 𝐶) = ((A · 𝐶) + (B · 𝐶))

Theoremrecni 6837 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
A        A

Theoremreaddcli 6838 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
A     &   B        (A + B)

Theoremremulcli 6839 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
A     &   B        (A · B)

Theorem1red 6840 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(φ → 1 ℝ)

Theorem1cnd 6841 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(φ → 1 ℂ)

Theoremmulid1d 6842 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 1) = A)

Theoremmulid2d 6843 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (1 · A) = A)

(φA ℂ)    &   (φB ℂ)       (φ → (A + B) ℂ)

Theoremmulcld 6845 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · B) ℂ)

Theoremmulcomd 6846 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · B) = (B · A))

(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) + 𝐶) = (A + (B + 𝐶)))

Theoremmulassd 6848 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 6849 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 6850 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)))

Theoremrecnd 6851 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(φA ℝ)       (φA ℂ)

Theoremreaddcld 6852 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A + B) ℝ)

Theoremremulcld 6853 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 6854 Plus infinity.
class +∞

Syntaxcmnf 6855 Minus infinity.
class -∞

Syntaxcxr 6856 The set of extended reals (includes plus and minus infinity).
class *

Syntaxclt 6857 'Less than' predicate (extended to include the extended reals).
class <

Syntaxcle 6858 Extend wff notation to include the 'less than or equal to' relation.
class

Definitiondf-pnf 6859 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 6860). 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 6864 and mnfnre 6865, 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 6860 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 6865). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞

Definitiondf-xr 6861 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 6862* 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 y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))

Definitiondf-le 6863 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )

Theorempnfnre 6864 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ

Theoremmnfnre 6865 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ

Theoremressxr 6866 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*

Theoremrexpssxrxp 6867 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 6868 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(A ℝ → A *)

Theorem0xr 6869 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 *

Theoremrenepnf 6870 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(A ℝ → A ≠ +∞)

Theoremrenemnf 6871 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(A ℝ → A ≠ -∞)

Theoremrexrd 6872 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA *)

Theoremrenepnfd 6873 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≠ +∞)

Theoremrenemnfd 6874 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≠ -∞)

Theoremrexri 6875 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
A        A *

Theoremrenfdisj 6876 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅

Theoremltrelxr 6877 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)

Theoremltrel 6878 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <

Theoremlerelxr 6879 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)

Theoremlerel 6880 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤

Theoremxrlenlt 6881 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
((A * B *) → (AB ↔ ¬ B < A))

Theoremltxrlt 6882 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 B ℝ) → (A < BA < B))

3.2.3  Restate the ordering postulates with extended real "less than"

Theoremaxltirr 6883 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6795 with ordering on the extended reals. New proofs should use ltnr 6892 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(A ℝ → ¬ A < A)

Theoremaxltwlin 6884 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6796 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
((A B 𝐶 ℝ) → (A < B → (A < 𝐶 𝐶 < B)))

Theoremaxlttrn 6885 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6797 with ordering on the extended reals. New proofs should use lttr 6889 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))

Theoremaxltadd 6886 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 6799 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((A B 𝐶 ℝ) → (A < B → (𝐶 + A) < (𝐶 + B)))

Theoremaxapti 6887 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6798 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((A B ¬ (A < B B < A)) → A = B)

Theoremaxmulgt0 6888 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 6800 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((A B ℝ) → ((0 < A 0 < B) → 0 < (A · B)))

3.2.4  Ordering on reals

Theoremlttr 6889 Alias for axlttrn 6885, for naming consistency with lttri 6919. New proofs should generally use this instead of ax-pre-lttrn 6797. (Contributed by NM, 10-Mar-2008.)
((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))

Theoremmulgt0 6890 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
(((A 0 < A) (B 0 < B)) → 0 < (A · B))

Theoremlenlt 6891 'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)
((A B ℝ) → (AB ↔ ¬ B < A))

Theoremltnr 6892 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
(A ℝ → ¬ A < A)

Theoremltso 6893 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ

Theoremgtso 6894 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ

Theoremlttri3 6895 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
((A B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))

Theoremletri3 6896 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
((A B ℝ) → (A = B ↔ (AB BA)))

Theoremltleletr 6897 Transitive law, weaker form of (A < B B𝐶) → A < 𝐶. (Contributed by AV, 14-Oct-2018.)
((A B 𝐶 ℝ) → ((A < B B𝐶) → A𝐶))

Theoremletr 6898 Transitive law. (Contributed by NM, 12-Nov-1999.)
((A B 𝐶 ℝ) → ((AB B𝐶) → A𝐶))

Theoremleid 6899 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
(A ℝ → AA)

Theoremltne 6900 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
((A A < B) → BA)

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