HomeHome Metamath Proof Explorer
Theorem List (p. 280 of 480)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30435)
  Hilbert Space Explorer  Hilbert Space Explorer
(30436-31958)
  Users' Mathboxes  Users' Mathboxes
(31959-47941)
 

Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecsex 27901* A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 15-Mar-2025.)
((𝐴 ∈ No ∧ 𝐴 β‰  0s ) β†’ βˆƒπ‘₯ ∈ No (𝐴 Β·s π‘₯) = 1s )
 
Theoremrecsexd 27902* A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐴 β‰  0s )    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ No (𝐴 Β·s π‘₯) = 1s )
 
Theoremdivsmul 27903 Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ (𝐢 ∈ No ∧ 𝐢 β‰  0s )) β†’ ((𝐴 /su 𝐢) = 𝐡 ↔ (𝐢 Β·s 𝐡) = 𝐴))
 
Theoremdivsmuld 27904 Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 𝐢 β‰  0s )    β‡’   (πœ‘ β†’ ((𝐴 /su 𝐢) = 𝐡 ↔ (𝐢 Β·s 𝐡) = 𝐴))
 
Theoremdivscl 27905 Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐡 β‰  0s ) β†’ (𝐴 /su 𝐡) ∈ No )
 
Theoremdivscld 27906 Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐡 β‰  0s )    β‡’   (πœ‘ β†’ (𝐴 /su 𝐡) ∈ No )
 
Theoremdivscan2d 27907 A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐡 β‰  0s )    β‡’   (πœ‘ β†’ (𝐡 Β·s (𝐴 /su 𝐡)) = 𝐴)
 
Theoremdivscan1d 27908 A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐡 β‰  0s )    β‡’   (πœ‘ β†’ ((𝐴 /su 𝐡) Β·s 𝐡) = 𝐴)
 
Theoremsltdivmuld 27909 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 0s <s 𝐢)    β‡’   (πœ‘ β†’ ((𝐴 /su 𝐢) <s 𝐡 ↔ 𝐴 <s (𝐢 Β·s 𝐡)))
 
Theoremsltdivmul2d 27910 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 0s <s 𝐢)    β‡’   (πœ‘ β†’ ((𝐴 /su 𝐢) <s 𝐡 ↔ 𝐴 <s (𝐡 Β·s 𝐢)))
 
Theoremsltmuldivd 27911 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 0s <s 𝐢)    β‡’   (πœ‘ β†’ ((𝐴 Β·s 𝐢) <s 𝐡 ↔ 𝐴 <s (𝐡 /su 𝐢)))
 
Theoremsltmuldiv2d 27912 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 0s <s 𝐢)    β‡’   (πœ‘ β†’ ((𝐢 Β·s 𝐴) <s 𝐡 ↔ 𝐴 <s (𝐡 /su 𝐢)))
 
Theoremdivsassd 27913 An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ 𝐢 β‰  0s )    β‡’   (πœ‘ β†’ ((𝐴 Β·s 𝐡) /su 𝐢) = (𝐴 Β·s (𝐡 /su 𝐢)))
 
15.6  Subsystems of surreals
 
15.6.1  Ordinal numbers
 
Syntaxcons 27914 Declare the syntax for surreal ordinals.
class Ons
 
Definitiondf-ons 27915 Define the surreal ordinals. These are the maximum members of each generation of surreals. Variant of definition from [Conway] p. 27. (Contributed by Scott Fenton, 18-Mar-2025.)
Ons = {π‘₯ ∈ No ∣ ( R β€˜π‘₯) = βˆ…}
 
Theoremelons 27916 Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025.)
(𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R β€˜π΄) = βˆ…))
 
Theoremonssno 27917 The surreal ordinals are a subclass of the surreals. (Contributed by Scott Fenton, 18-Mar-2025.)
Ons βŠ† No
 
Theoremonsno 27918 A surreal ordinal is a surreal. (Contributed by Scott Fenton, 18-Mar-2025.)
(𝐴 ∈ Ons β†’ 𝐴 ∈ No )
 
Theorem0ons 27919 Surreal zero is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
0s ∈ Ons
 
Theorem1ons 27920 Surreal one is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
1s ∈ Ons
 
Theoremelons2 27921* A surreal is ordinal iff it is the cut of some set of surreals and the empty set. Definition from [Conway] p. 27. (Contributed by Scott Fenton, 19-Mar-2025.)
(𝐴 ∈ Ons ↔ βˆƒπ‘Ž ∈ 𝒫 No 𝐴 = (π‘Ž |s βˆ…))
 
Theoremelons2d 27922 The cut of any set of surreals and the empty set is a surreal ordinal. (Contributed by Scott Fenton, 19-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 βŠ† No )    &   (πœ‘ β†’ 𝑋 = (𝐴 |s βˆ…))    β‡’   (πœ‘ β†’ 𝑋 ∈ Ons)
 
Theoremsltonold 27923* The class of ordinals less than any surreal is a subset of that surreal's old set. (Contributed by Scott Fenton, 22-Mar-2025.)
(𝐴 ∈ No β†’ {π‘₯ ∈ Ons ∣ π‘₯ <s 𝐴} βŠ† ( O β€˜( bday β€˜π΄)))
 
Theoremsltonex 27924* The class of ordinals less than any particular surreal is a set. Theorem 14 of [Conway] p. 27. (Contributed by Scott Fenton, 22-Mar-2025.)
(𝐴 ∈ No β†’ {π‘₯ ∈ Ons ∣ π‘₯ <s 𝐴} ∈ V)
 
Theoremonscutleft 27925 A surreal ordinal is equal to the cut of its left set and the empty set. (Contributed by Scott Fenton, 29-Mar-2025.)
(𝐴 ∈ Ons β†’ 𝐴 = (( L β€˜π΄) |s βˆ…))
 
15.6.2  Natural numbers
 
Syntaxcnn0s 27926 Declare the syntax for surreal non-negative integers.
class β„•0s
 
Syntaxcnns 27927 Declare the syntax for surreal positive integers.
class β„•s
 
Definitiondf-n0s 27928 Define the set of non-negative surreal integers. This set behaves similarly to Ο‰ and β„•0, but it is a set of surreal numbers. Like those two sets, it satisfies the Peano axioms and is closed under (surreal) addition and multiplication. Compare df-nn 12218. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•0s = (rec((π‘₯ ∈ V ↦ (π‘₯ +s 1s )), 0s ) β€œ Ο‰)
 
Definitiondf-nns 27929 Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•s = (β„•0s βˆ– { 0s })
 
Theoremn0sex 27930 The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•0s ∈ V
 
Theoremnnsex 27931 The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•s ∈ V
 
Theorempeano5n0s 27932* Peano's inductive postulate for non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
(( 0s ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘₯ +s 1s ) ∈ 𝐴) β†’ β„•0s βŠ† 𝐴)
 
Theoremn0ssno 27933 The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•0s βŠ† No
 
Theoremnnssn0s 27934 The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•s βŠ† β„•0s
 
Theoremnnssno 27935 The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•s βŠ† No
 
Theorem0n0s 27936 Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
0s ∈ β„•0s
 
Theorempeano2n0s 27937 Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝐴 ∈ β„•0s β†’ (𝐴 +s 1s ) ∈ β„•0s)
 
Theoremdfn0s2 27938* Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
β„•0s = ∩ {π‘₯ ∣ ( 0s ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 +s 1s ) ∈ π‘₯)}
 
Theoremn0sind 27939* Principle of Mathematical Induction (inference schema). Compare nnind 12235 and finds 7892. (Contributed by Scott Fenton, 17-Mar-2025.)
(π‘₯ = 0s β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ’))    &   (π‘₯ = (𝑦 +s 1s ) β†’ (πœ‘ ↔ πœƒ))    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))    &   πœ“    &   (𝑦 ∈ β„•0s β†’ (πœ’ β†’ πœƒ))    β‡’   (𝐴 ∈ β„•0s β†’ 𝜏)
 
Theoremn0scut 27940 A cut form for surreal naturals. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ β„•0s β†’ 𝐴 = ({(𝐴 -s 1s )} |s βˆ…))
 
Theoremn0ons 27941 A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ β„•0s β†’ 𝐴 ∈ Ons)
 
PART 16  ELEMENTARY GEOMETRY

This part develops elementary geometry based on Tarski's axioms, following [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the "points". It has two primitive notions, the ternary predicate of "betweenness" and the quaternary predicate of "congruence". To adapt this theory to the framework of set.mm, and to be able to talk of *a* Tarski structure as a space satisfying the given axioms, we use the following definition, stated informally:

A Tarski structure 𝑓 is a set (of points) (Baseβ€˜π‘“) together with functions (Itvβ€˜π‘“) and (distβ€˜π‘“) on ((Baseβ€˜π‘“) Γ— (Baseβ€˜π‘“)) satisfying certain axioms (given in Definitions df-trkg 27968 et sequentes). This allows to treat a Tarski structure as a special kind of extensible structure (see df-struct 17085).

The translation to and from Tarski's treatment is as follows (given, again, informally).

Suppose that one is given an extensible structure 𝑓. One defines a betweenness ternary predicate Btw by positing that, for any π‘₯, 𝑦, 𝑧 ∈ (Baseβ€˜π‘“), one has "Btw π‘₯𝑦𝑧 " if and only if 𝑦 ∈ π‘₯(Itvβ€˜π‘“)𝑧, and a congruence quaternary predicate Congr by positing that, for any π‘₯, 𝑦, 𝑧, 𝑑 ∈ (Baseβ€˜π‘“), one has "Congr π‘₯𝑦𝑧𝑑 " if and only if π‘₯(distβ€˜π‘“)𝑦 = 𝑧(distβ€˜π‘“)𝑑. It is easy to check that if 𝑓 satisfies our Tarski axioms, then Btw and Congr satisfy Tarski's Tarski axioms when (Baseβ€˜π‘“) is interpreted as the universe of discourse.

Conversely, suppose that one is given a set π‘Ž, a ternary predicate Btw, and a quaternary predicate Congr. One defines the extensible structure 𝑓 such that (Baseβ€˜π‘“) is π‘Ž, and (Itvβ€˜π‘“) is the function which associates with each ⟨π‘₯, π‘¦βŸ© ∈ (π‘Ž Γ— π‘Ž) the set of points 𝑧 ∈ π‘Ž such that "Btw π‘₯𝑧𝑦", and (distβ€˜π‘“) is the function which associates with each ⟨π‘₯, π‘¦βŸ© ∈ (π‘Ž Γ— π‘Ž) the set of ordered pairs βŸ¨π‘§, π‘‘βŸ© ∈ (π‘Ž Γ— π‘Ž) such that "Congr π‘₯𝑦𝑧𝑑". It is easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when π‘Ž is interpreted as the universe of discourse, then 𝑓 satisfies our Tarski axioms.

We intentionally choose to represent congruence (without loss of generality) as π‘₯(distβ€˜π‘“)𝑦 = 𝑧(distβ€˜π‘“)𝑑 instead of "Congr π‘₯𝑦𝑧𝑑", as it is more convenient. It is always possible to define dist for any particular geometry to produce equal results when conguence is desired, and in many cases there is an obvious interpretation of "distance" between two points that can be useful in other situations. Encoding congruence as an equality of distances makes it easier to use these theorems in cases where there is a preferred distance function. We prove that representing a congruence relationship using a distance in the form π‘₯(distβ€˜π‘“)𝑦 = 𝑧(distβ€˜π‘“)𝑑 causes no loss of generality in tgjustc1 27990 and tgjustc2 27991, which in turn are supported by tgjustf 27988 and tgjustr 27989.

A similar representation of congruence (using a "distance" function) is used in Axiom A1 of [Beeson2016] p. 5, which discusses how a large number of formalized proofs were found in Tarskian Geometry using OTTER. Their detailed proofs in Tarski Geometry, along with other information, are available at https://www.michaelbeeson.com/research/FormalTarski/ 27989.

Most theorems are in deduction form, as this is a very general, simple, and convenient format to use in Metamath. An assertion in deduction form can be easily converted into an assertion in inference form (removing the antecedents πœ‘ β†’) by insert a ⊀ β†’ in each hypothesis, using a1i 11, then using mptru 1547 to remove the final ⊀ β†’ prefix. In some cases we represent, without loss of generality, an implication antecedent in [Schwabhauser] as a hypothesis. The implication can be retrieved from the by using simpr 484, the theorem as stated, and ex 412.

For descriptions of individual axioms, we refer to the specific definitions below. A particular feature of Tarski's axioms is modularity, so by using various subsets of the set of axioms, we can define the classes of "absolute dimensionless Tarski structures" (df-trkg 27968), of "Euclidean dimensionless Tarski structures" (df-trkge 27966) and of "Tarski structures of dimension no less than N" (df-trkgld 27967).

In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 28323 and iscgra 28324). To maintain its simplicity, in this system congruence between shapes (a finite sequence of points) is the case where corresponding segments between all corresponding points are congruent. This includes triangles (a shape of 3 distinct points). Note that this definition has no direct regard for angles. For more details and rationale, see df-cgrg 28026.

The first section is devoted to the definitions of these various structures. The second section ("Tarskian geometry") develops the synthetic treatment of geometry. The remaining sections prove that real Euclidean spaces and complex Hilbert spaces, with intended interpretations, are Euclidean Tarski structures.

Most of the work in this part is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. See also the credits in the comment of each statement.

 
16.1  Definition and Tarski's Axioms of Geometry
 
Syntaxcstrkg 27942 Extends class notation with the class of Tarski geometries.
class TarskiG
 
Syntaxcstrkgc 27943 Extends class notation with the class of geometries fulfilling the congruence axioms.
class TarskiGC
 
Syntaxcstrkgb 27944 Extends class notation with the class of geometries fulfilling the betweenness axioms.
class TarskiGB
 
Syntaxcstrkgcb 27945 Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms.
class TarskiGCB
 
Syntaxcstrkgld 27946 Extends class notation with the relation for geometries fulfilling the lower dimension axioms.
class DimTarskiGβ‰₯
 
Syntaxcstrkge 27947 Extends class notation with the class of geometries fulfilling Euclid's axiom.
class TarskiGE
 
Syntaxcitv 27948 Declare the syntax for the Interval (segment) index extractor.
class Itv
 
Syntaxclng 27949 Declare the syntax for the Line function.
class LineG
 
Definitiondf-itv 27950 Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent form itvid 27954 instead. (New usage is discouraged.)
Itv = Slot 16
 
Definitiondf-lng 27951 Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.) Use its index-independent form lngid 27955 instead. (New usage is discouraged.)
LineG = Slot 17
 
Theoremitvndx 27952 Index value of the Interval (segment) slot. Use ndxarg 17134. (Contributed by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.)
(Itvβ€˜ndx) = 16
 
Theoremlngndx 27953 Index value of the "line" slot. Use ndxarg 17134. (Contributed by Thierry Arnoux, 27-Mar-2019.) (New usage is discouraged.)
(LineGβ€˜ndx) = 17
 
Theoremitvid 27954 Utility theorem: index-independent form of df-itv 27950. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv = Slot (Itvβ€˜ndx)
 
Theoremlngid 27955 Utility theorem: index-independent form of df-lng 27951. (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG = Slot (LineGβ€˜ndx)
 
Theoremslotsinbpsd 27956 The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 28392 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((Itvβ€˜ndx) β‰  (Baseβ€˜ndx) ∧ (Itvβ€˜ndx) β‰  (+gβ€˜ndx)) ∧ ((Itvβ€˜ndx) β‰  ( ·𝑠 β€˜ndx) ∧ (Itvβ€˜ndx) β‰  (distβ€˜ndx)))
 
Theoremslotslnbpsd 27957 The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 28392 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((LineGβ€˜ndx) β‰  (Baseβ€˜ndx) ∧ (LineGβ€˜ndx) β‰  (+gβ€˜ndx)) ∧ ((LineGβ€˜ndx) β‰  ( ·𝑠 β€˜ndx) ∧ (LineGβ€˜ndx) β‰  (distβ€˜ndx)))
 
Theoremlngndxnitvndx 27958 The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 28390. (Contributed by AV, 9-Nov-2024.)
(LineGβ€˜ndx) β‰  (Itvβ€˜ndx)
 
Theoremtrkgstr 27959 Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
π‘Š = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(distβ€˜ndx), 𝐷⟩, ⟨(Itvβ€˜ndx), 𝐼⟩}    β‡’   π‘Š Struct ⟨1, 16⟩
 
Theoremtrkgbas 27960 The base set of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
π‘Š = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(distβ€˜ndx), 𝐷⟩, ⟨(Itvβ€˜ndx), 𝐼⟩}    β‡’   (π‘ˆ ∈ 𝑉 β†’ π‘ˆ = (Baseβ€˜π‘Š))
 
Theoremtrkgdist 27961 The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
π‘Š = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(distβ€˜ndx), 𝐷⟩, ⟨(Itvβ€˜ndx), 𝐼⟩}    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐷 = (distβ€˜π‘Š))
 
Theoremtrkgitv 27962 The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
π‘Š = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(distβ€˜ndx), 𝐷⟩, ⟨(Itvβ€˜ndx), 𝐼⟩}    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐼 = (Itvβ€˜π‘Š))
 
Definitiondf-trkgc 27963* Define the class of geometries fulfilling the congruence axioms of reflexivity, identity and transitivity. These are axioms A1 to A3 of [Schwabhauser] p. 10. With our distance based notation for congruence, transitivity of congruence boils down to transitivity of equality and is already given by eqtr 2754, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
TarskiGC = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))}
 
Definitiondf-trkgb 27964* Define the class of geometries fulfilling the 3 betweenness axioms in Tarski's Axiomatization of Geometry: identity, Axiom A6 of [Schwabhauser] p. 11, axiom of Pasch, Axiom A7 of [Schwabhauser] p. 12, and continuity, Axiom A11 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 24-Aug-2017.)
TarskiGB = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (𝑦 ∈ (π‘₯𝑖π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 βˆ€π‘’ ∈ 𝑝 βˆ€π‘£ ∈ 𝑝 ((𝑒 ∈ (π‘₯𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑝 (π‘Ž ∈ (𝑒𝑖𝑦) ∧ π‘Ž ∈ (𝑣𝑖π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 π‘βˆ€π‘‘ ∈ 𝒫 𝑝(βˆƒπ‘Ž ∈ 𝑝 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Žπ‘–π‘¦) β†’ βˆƒπ‘ ∈ 𝑝 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯𝑖𝑦)))}
 
Definitiondf-trkgcb 27965* Define the class of geometries fulfilling the five segment axiom, Axiom A5 of [Schwabhauser] p. 11, and segment construction axiom, Axiom A4 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiGCB = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑][(Itvβ€˜π‘“) / 𝑖](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 βˆ€π‘’ ∈ 𝑝 βˆ€π‘Ž ∈ 𝑝 βˆ€π‘ ∈ 𝑝 βˆ€π‘ ∈ 𝑝 βˆ€π‘£ ∈ 𝑝 (((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯𝑖𝑧) ∧ 𝑏 ∈ (π‘Žπ‘–π‘)) ∧ (((π‘₯𝑑𝑦) = (π‘Žπ‘‘π‘) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((π‘₯𝑑𝑒) = (π‘Žπ‘‘π‘£) ∧ (𝑦𝑑𝑒) = (𝑏𝑑𝑣)))) β†’ (𝑧𝑑𝑒) = (𝑐𝑑𝑣)) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘Ž ∈ 𝑝 βˆ€π‘ ∈ 𝑝 βˆƒπ‘§ ∈ 𝑝 (𝑦 ∈ (π‘₯𝑖𝑧) ∧ (𝑦𝑑𝑧) = (π‘Žπ‘‘π‘)))}
 
Definitiondf-trkge 27966* Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiGE = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖]βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 βˆ€π‘’ ∈ 𝑝 βˆ€π‘£ ∈ 𝑝 ((𝑒 ∈ (π‘₯𝑖𝑣) ∧ 𝑒 ∈ (𝑦𝑖𝑧) ∧ π‘₯ β‰  𝑒) β†’ βˆƒπ‘Ž ∈ 𝑝 βˆƒπ‘ ∈ 𝑝 (𝑦 ∈ (π‘₯π‘–π‘Ž) ∧ 𝑧 ∈ (π‘₯𝑖𝑏) ∧ 𝑣 ∈ (π‘Žπ‘–π‘)))}
 
Definitiondf-trkgld 27967* Define the class of geometries fulfilling the lower dimension axiom for dimension 𝑛. For such geometries, there are three non-colinear points that are equidistant from 𝑛 βˆ’ 1 distinct points. Derived from remarks in Tarski's System of Geometry, Alfred Tarski and Steven Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.) (Revised by Thierry Arnoux, 23-Nov-2019.)
DimTarskiGβ‰₯ = {βŸ¨π‘”, π‘›βŸ© ∣ [(Baseβ€˜π‘”) / 𝑝][(distβ€˜π‘”) / 𝑑][(Itvβ€˜π‘”) / 𝑖]βˆƒπ‘“(𝑓:(1..^𝑛)–1-1→𝑝 ∧ βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 βˆƒπ‘§ ∈ 𝑝 (βˆ€π‘— ∈ (2..^𝑛)(((π‘“β€˜1)𝑑π‘₯) = ((π‘“β€˜π‘—)𝑑π‘₯) ∧ ((π‘“β€˜1)𝑑𝑦) = ((π‘“β€˜π‘—)𝑑𝑦) ∧ ((π‘“β€˜1)𝑑𝑧) = ((π‘“β€˜π‘—)𝑑𝑧)) ∧ Β¬ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))))}
 
Definitiondf-trkg 27968* Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence, (π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑀) where βˆ’ = (distβ€˜π‘Š)
  • for betweenness, 𝑦 ∈ (π‘₯𝐼𝑧), where 𝐼 = (Itvβ€˜π‘Š)
With this definition, the axiom A2 is actually equivalent to the transitivity of equality, eqtrd 2771.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (β—‘DimTarskiGβ‰₯ β€œ {2})
  • A9 Upper dimension axiom (V βˆ– (β—‘DimTarskiGβ‰₯ β€œ {3}))
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms DimTarskiGβ‰₯
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})}))
 
Theoremistrkgc 27969* Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦))))
 
Theoremistrkgb 27970* Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (𝑦 ∈ (π‘₯𝐼π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 π‘ƒβˆ€π‘‘ ∈ 𝒫 𝑃(βˆƒπ‘Ž ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘ŽπΌπ‘¦) β†’ βˆƒπ‘ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯𝐼𝑦)))))
 
Theoremistrkgcb 27971* Property of being a Tarski geometry - congruence and betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 βˆ€π‘ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 (((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯𝐼𝑧) ∧ 𝑏 ∈ (π‘ŽπΌπ‘)) ∧ (((π‘₯ βˆ’ 𝑦) = (π‘Ž βˆ’ 𝑏) ∧ (𝑦 βˆ’ 𝑧) = (𝑏 βˆ’ 𝑐)) ∧ ((π‘₯ βˆ’ 𝑒) = (π‘Ž βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑏 βˆ’ 𝑣)))) β†’ (𝑧 βˆ’ 𝑒) = (𝑐 βˆ’ 𝑣)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘Ž ∈ 𝑃 βˆ€π‘ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 (𝑦 ∈ (π‘₯𝐼𝑧) ∧ (𝑦 βˆ’ 𝑧) = (π‘Ž βˆ’ 𝑏)))))
 
Theoremistrkge 27972* Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ TarskiGE ↔ (𝐺 ∈ V ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑣) ∧ 𝑒 ∈ (𝑦𝐼𝑧) ∧ π‘₯ β‰  𝑒) β†’ βˆƒπ‘Ž ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 (𝑦 ∈ (π‘₯πΌπ‘Ž) ∧ 𝑧 ∈ (π‘₯𝐼𝑏) ∧ 𝑣 ∈ (π‘ŽπΌπ‘)))))
 
Theoremistrkgl 27973* Building lines from the segment property. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineGβ€˜πΊ) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})))
 
Theoremistrkgld 27974* Property of fulfilling the lower dimension 𝑁 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝐺DimTarskiGβ‰₯𝑁 ↔ βˆƒπ‘“(𝑓:(1..^𝑁)–1-1→𝑃 ∧ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 (βˆ€π‘— ∈ (2..^𝑁)(((π‘“β€˜1) βˆ’ π‘₯) = ((π‘“β€˜π‘—) βˆ’ π‘₯) ∧ ((π‘“β€˜1) βˆ’ 𝑦) = ((π‘“β€˜π‘—) βˆ’ 𝑦) ∧ ((π‘“β€˜1) βˆ’ 𝑧) = ((π‘“β€˜π‘—) βˆ’ 𝑧)) ∧ Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
 
Theoremistrkg2ld 27975* Property of fulfilling the lower dimension 2 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ (𝐺DimTarskiGβ‰₯2 ↔ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))
 
Theoremistrkg3ld 27976* Property of fulfilling the lower dimension 3 axiom. (Contributed by Thierry Arnoux, 12-Jul-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ (𝐺DimTarskiGβ‰₯3 ↔ βˆƒπ‘’ ∈ 𝑃 βˆƒπ‘£ ∈ 𝑃 (𝑒 β‰  𝑣 ∧ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 (((𝑒 βˆ’ π‘₯) = (𝑣 βˆ’ π‘₯) ∧ (𝑒 βˆ’ 𝑦) = (𝑣 βˆ’ 𝑦) ∧ (𝑒 βˆ’ 𝑧) = (𝑣 βˆ’ 𝑧)) ∧ Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
 
Theoremaxtgcgrrflx 27977 Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) = (π‘Œ βˆ’ 𝑋))
 
Theoremaxtgcgrid 27978 Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑍 ∈ 𝑃)    &   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) = (𝑍 βˆ’ 𝑍))    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremaxtgsegcon 27979* Axiom of segment construction, Axiom A4 of [Schwabhauser] p. 11. As discussed in Axiom 4 of [Tarski1999] p. 178, "The intuitive content [is that] given any line segment 𝐴𝐡, one can construct a line segment congruent to it, starting at any point π‘Œ and going in the direction of any ray containing π‘Œ. The ray is determined by the point π‘Œ and a second point 𝑋, the endpoint of the ray. The other endpoint of the line segment to be constructed is just the point 𝑧 whose existence is asserted." (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ βˆƒπ‘§ ∈ 𝑃 (π‘Œ ∈ (𝑋𝐼𝑧) ∧ (π‘Œ βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡)))
 
Theoremaxtg5seg 27980 Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles π‘‹π‘π‘ˆ and 𝐴𝐢𝑉, a point π‘Œ on 𝑋𝑍, and a point 𝐡 on 𝐴𝐢. If all corresponding line segments except for π‘π‘ˆ and 𝐢𝑉 are congruent ( i.e., π‘‹π‘Œ ∼ 𝐴𝐡, π‘Œπ‘ ∼ 𝐡𝐢, π‘‹π‘ˆ ∼ 𝐴𝑉, and π‘Œπ‘ˆ ∼ 𝐡𝑉), then π‘π‘ˆ and 𝐢𝑉 are also congruent. As noted in Axiom 5 of [Tarski1999] p. 178, "this axiom is similar in character to the well-known theorems of Euclidean geometry that allow one to conclude, from hypotheses about the congruence of certain corresponding sides and angles in two triangles, the congruence of other corresponding sides and angles." (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑍 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑃)    &   (πœ‘ β†’ 𝑉 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋 β‰  π‘Œ)    &   (πœ‘ β†’ π‘Œ ∈ (𝑋𝐼𝑍))    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))    &   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) = (𝐴 βˆ’ 𝐡))    &   (πœ‘ β†’ (π‘Œ βˆ’ 𝑍) = (𝐡 βˆ’ 𝐢))    &   (πœ‘ β†’ (𝑋 βˆ’ π‘ˆ) = (𝐴 βˆ’ 𝑉))    &   (πœ‘ β†’ (π‘Œ βˆ’ π‘ˆ) = (𝐡 βˆ’ 𝑉))    β‡’   (πœ‘ β†’ (𝑍 βˆ’ π‘ˆ) = (𝐢 βˆ’ 𝑉))
 
Theoremaxtgbtwnid 27981 Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ (𝑋𝐼𝑋))    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremaxtgpasch 27982* Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle π‘‹π‘Œπ‘, point π‘ˆ in segment 𝑋𝑍, and point 𝑉 in segment π‘Œπ‘, there exists a point π‘Ž on both the segment π‘ˆπ‘Œ and the segment 𝑉𝑋. This axiom is essentially a subset of the general Pasch axiom. The general Pasch axiom asserts that on a plane "a line intersecting a triangle in one of its sides, and not intersecting any of the vertices, must intersect one of the other two sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The (general) Pasch axiom was used implicitly by Euclid, but never stated; Moritz Pasch discovered its omission in 1882. As noted in the Metamath book, this means that the omission of Pasch's axiom from Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm is included as an axiom; the "outer" form of the Pasch axiom can be proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑍 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑃)    &   (πœ‘ β†’ 𝑉 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ (𝑋𝐼𝑍))    &   (πœ‘ β†’ 𝑉 ∈ (π‘ŒπΌπ‘))    β‡’   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))
 
Theoremaxtgcont1 27983* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom (scheme) asserts that any two sets 𝑆 and 𝑇 (of points) such that the elements of 𝑆 precede the elements of 𝑇 with respect to some point π‘Ž (that is, π‘₯ is between π‘Ž and 𝑦 whenever π‘₯ is in 𝑋 and 𝑦 is in π‘Œ) are separated by some point 𝑏; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑆 βŠ† 𝑃)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑃)    β‡’   (πœ‘ β†’ (βˆƒπ‘Ž ∈ 𝑃 βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑇 π‘₯ ∈ (π‘ŽπΌπ‘¦) β†’ βˆƒπ‘ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑇 𝑏 ∈ (π‘₯𝐼𝑦)))
 
Theoremaxtgcont 27984* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 27983. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑆 βŠ† 𝑃)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   ((πœ‘ ∧ 𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) β†’ 𝑒 ∈ (𝐴𝐼𝑣))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑇 𝑏 ∈ (π‘₯𝐼𝑦))
 
Theoremaxtglowdim2 27985* Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)))
 
Theoremaxtgupdim2 27986 Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. Three points 𝑋, π‘Œ and 𝑍 equidistant to two given two points π‘ˆ and 𝑉 must be colinear. (Contributed by Thierry Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑍 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑃)    &   (πœ‘ β†’ 𝑉 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ β‰  𝑉)    &   (πœ‘ β†’ (π‘ˆ βˆ’ 𝑋) = (𝑉 βˆ’ 𝑋))    &   (πœ‘ β†’ (π‘ˆ βˆ’ π‘Œ) = (𝑉 βˆ’ π‘Œ))    &   (πœ‘ β†’ (π‘ˆ βˆ’ 𝑍) = (𝑉 βˆ’ 𝑍))    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝐺DimTarskiGβ‰₯3)    β‡’   (πœ‘ β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))
 
Theoremaxtgeucl 27987* Euclid's Axiom. Axiom A10 of [Schwabhauser] p. 13. This is equivalent to Euclid's parallel postulate when combined with other axioms. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiGE)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑍 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑃)    &   (πœ‘ β†’ 𝑉 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ (𝑋𝐼𝑉))    &   (πœ‘ β†’ π‘ˆ ∈ (π‘ŒπΌπ‘))    &   (πœ‘ β†’ 𝑋 β‰  π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 (π‘Œ ∈ (π‘‹πΌπ‘Ž) ∧ 𝑍 ∈ (𝑋𝐼𝑏) ∧ 𝑉 ∈ (π‘ŽπΌπ‘)))
 
16.1.1  Justification for the congruence notation
 
Theoremtgjustf 27988* Given any function 𝐹, equality of the image by 𝐹 is an equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.)
(𝐴 ∈ 𝑉 β†’ βˆƒπ‘Ÿ(π‘Ÿ Er 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯π‘Ÿπ‘¦ ↔ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))))
 
Theoremtgjustr 27989* Given any equivalence relation 𝑅, one can define a function 𝑓 such that all elements of an equivalence classe of 𝑅 have the same image by 𝑓. (Contributed by Thierry Arnoux, 25-Jan-2023.)
((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 ↔ (π‘“β€˜π‘₯) = (π‘“β€˜π‘¦))))
 
Theoremtgjustc1 27990* A justification for using distance equality instead of the textbook relation on pairs of points for congruence. (Contributed by Thierry Arnoux, 29-Jan-2023.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    β‡’   βˆƒπ‘Ÿ(π‘Ÿ Er (𝑃 Γ— 𝑃) ∧ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘ŸβŸ¨π‘¦, π‘§βŸ© ↔ (𝑀 βˆ’ π‘₯) = (𝑦 βˆ’ 𝑧)))
 
Theoremtgjustc2 27991* A justification for using distance equality instead of the textbook relation on pairs of points for congruence. (Contributed by Thierry Arnoux, 29-Jan-2023.)
𝑃 = (Baseβ€˜πΊ)    &   π‘… Er (𝑃 Γ— 𝑃)    β‡’   βˆƒπ‘‘(𝑑 Fn (𝑃 Γ— 𝑃) ∧ βˆ€π‘€ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 (βŸ¨π‘€, π‘₯βŸ©π‘…βŸ¨π‘¦, π‘§βŸ© ↔ (𝑀𝑑π‘₯) = (𝑦𝑑𝑧)))
 
16.2  Tarskian Geometry
 
16.2.1  Congruence
 
Theoremtgcgrcomimp 27992 Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    β‡’   (πœ‘ β†’ ((𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐢)))
 
Theoremtgcgrcomr 27993 Congruence commutes on the RHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐢))
 
Theoremtgcgrcoml 27994 Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    β‡’   (πœ‘ β†’ (𝐡 βˆ’ 𝐴) = (𝐢 βˆ’ 𝐷))
 
Theoremtgcgrcomlr 27995 Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    β‡’   (πœ‘ β†’ (𝐡 βˆ’ 𝐴) = (𝐷 βˆ’ 𝐢))
 
Theoremtgcgreqb 27996 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    β‡’   (πœ‘ β†’ (𝐴 = 𝐡 ↔ 𝐢 = 𝐷))
 
Theoremtgcgreq 27997 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ 𝐢 = 𝐷)
 
Theoremtgcgrneq 27998 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐢 βˆ’ 𝐷))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    β‡’   (πœ‘ β†’ 𝐢 β‰  𝐷)
 
Theoremtgcgrtriv 27999 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐴) = (𝐡 βˆ’ 𝐡))
 
Theoremtgcgrextend 28000 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))    &   (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47941
  Copyright terms: Public domain < Previous  Next >