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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempntibndlem2 23901* Lemma for pntibnd 23903. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ                                                                              ψ ψ

Theorempntibndlem3 23902* Lemma for pntibnd 23903. Package up pntibndlem2 23901 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibnd 23903* Lemma for pnt 23924. Establish smallness of on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntlemd 23904 Lemma for pnt 23924. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is C^*, is c1, is λ, is c2, and is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;

Theorempntlemc 23905* Lemma for pnt 23924. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is α, is ε, and is K. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;

Theorempntlema 23906* Lemma for pnt 23924. Closure for the constants used in the proof. The mammoth expression is a number large enough to satisfy all the lower bounds needed for . For comparison with Equation 10.6.27 of [Shapiro], p. 434, is x2, is x1, is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemb 23907* Lemma for pnt 23924. Unpack all the lower bounds contained in , in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;               ;

Theorempntlemg 23908* Lemma for pnt 23924. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is j^* and is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemh 23909* Lemma for pnt 23924. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemn 23910* Lemma for pnt 23924. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemq 23911* Lemma for pntlemj 23913. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemr 23912* Lemma for pntlemj 23913. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemj 23913* Lemma for pnt 23924. The induction step. Using pntibnd 23903, we find an interval in which is sufficiently large and has a much smaller value, (instead of our original bound ). (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemi 23914* Lemma for pnt 23924. Eliminate some assumptions from pntlemj 23913. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                  ..^

Theorempntlemf 23915* Lemma for pnt 23924. Add up the pieces in pntlemi 23914 to get an estimate slightly better than the naive lower bound . (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                           ;

Theorempntlemk 23916* Lemma for pnt 23924. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemo 23917* Lemma for pnt 23924. Combine all the estimates to establish a smaller eventual bound on . (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntleme 23918* Lemma for pnt 23924. Package up pntlemo 23917 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlem3 23919* Lemma for pnt 23924. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ                                           ψ

Theorempntlemp 23920* Lemma for pnt 23924. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;

Theorempntleml 23921* Lemma for pnt 23924. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;               ψ

Theorempnt3 23922 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)
ψ

Theorempnt2 23923 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)

Theorempnt 23924 The Prime Number Theorem: the number of prime numbers less than tends asymptotically to as goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
π

14.4.13  Ostrowski's theorem

Theoremabvcxp 23925* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theorempadicfval 23926* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theorempadicval 23927* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremostth2lem1 23928* Lemma for ostth2 23947, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 23947. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, for any . (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremqrngbas 23929 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqdrng 23930 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng0 23931 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng1 23932 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngneg 23933 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngdiv 23934 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds        /r

Theoremqabvle 23935 By using induction on , we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremqabvexp 23936 Induct the product rule abvmul 17604 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostthlem1 23937* Lemma for ostth 23949. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostthlem2 23938* Lemma for ostth 23949. Refine ostthlem1 23937 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
flds        AbsVal

Theoremqabsabv 23939 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabv 23940* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvf 23941* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvcxp 23942* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostth1 23943* - Lemma for ostth 23949: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If is equal to on the primes, then by complete induction and the multiplicative property abvmul 17604 of the absolute value, is equal to on all the integers, and ostthlem1 23937 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem2 23944* Lemma for ostth2 23947. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem3 23945* Lemma for ostth2 23947. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem4 23946* Lemma for ostth2 23947. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2 23947* - Lemma for ostth 23949: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth3 23948* - Lemma for ostth 23949: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth 23949* Ostrowski's theorem, which classifies all absolute values on . Any such absolute value must either be the trivial absolute value , a constant exponent times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

PART 15  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 is a set (of points) together with functions Itv and on satisfying certain axioms (given in the definitions df-trkg 23975 et sequentes). This allows us to treat a Tarski structure as a special kind of extensible structure (see df-struct 14645).

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

Suppose that one is given an extensible structure . One defines a betweenness ternary predicate Btw by positing that, for any , one has "Btw " if and only if Itv, and a congruence quaternary predicate Congr by positing that, for any , one has "Congr " if and only if . It is easy to check that if satisfies our Tarski axioms, then Btw and Congr satisfy Tarski's Tarski axioms when is interpreted as the universe of discourse.

Conversely, suppose that one is given a set , a ternary predicate Btw and a quaternary predicate Congr. One defines the extensible structure such that is , and Itv is the function which associates with each the set of points such that "Btw ", and is the function which associates with each the set of ordered pairs such that "Congr ". It is easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when is interpreted as the universe of discourse, then satisfies our Tarski axioms.

We intentionally choose to represent congruence (without loss of generality) as instead of "Congr ", as it is more convenient. It is always possible to define 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. A similar representation 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 http://www.michaelbeeson.com/research/FormalTarski/.

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 23975), of "Euclidean dimensionless Tarski structures" (df-trkge 23972) and of "plane Euclidean Tarski structures" (df-trkg2d 23974).

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 the 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.

15.1  Definition and Tarski's Axioms of Geometry

Syntaxcstrkg 23950 Extends class notation with the class of Tarski geometries.
TarskiG

Syntaxcstrkgc 23951 Extends class notation with the class of geometries fulfilling the congruence axioms.
TarskiGC

Syntaxcstrkgb 23952 Extends class notation with the class of geometries fulfilling the betweenness axioms.
TarskiGB

Syntaxcstrkgcb 23953 Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms.
TarskiGCB

Syntaxcstrkgld 23954 Extends class notation with the relation for geometries fulfilling the lower dimension axioms.
DimTarskiG

Syntaxcstrkg2d 23955 Extends class notation with the class of geometries fulfilling the planarity axioms.
TarskiG2D

Syntaxcstrkge 23956 Extends class notation with the class of geometries fulfilling Euclid's axiom.
TarskiGE

Syntaxcitv 23957 Declare the syntax for the Interval (segment) index extractor.
Itv

Syntaxclng 23958 Declare the syntax for the Line function.
LineG

Definitiondf-itv 23959 Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv Slot ;

Definitiondf-lng 23960 Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG Slot ;

Theoremitvndx 23961 Index value of the Interval (segment) slot. Use ndxarg 14663 (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv ;

Theoremlngndx 23962 Index value of the "line" slot. Use ndxarg 14663 (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG ;

Theoremitvid 23963 Utility theorem: index-independent form of df-itv 23959. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv Slot Itv

Theoremlngid 23964 Utility theorem: index-independent form of df-lng 23960. (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG Slot LineG

Theoremtrkgstr 23965 Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv        Struct ;

Theoremtrkgbas 23966 The base set of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv

Theoremtrkgdist 23967 The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv

Theoremtrkgitv 23968 The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv        Itv

Definitiondf-trkgc 23969* 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 2483, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
TarskiGC

Definitiondf-trkgb 23970* 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 Itv

Definitiondf-trkgcb 23971* 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 Itv

Definitiondf-trkge 23972* Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiGE Itv

Definitiondf-trkgld 23973* Define the class of geometries fulfilling the lower dimension axiom for dimension . For such geometries, there are three non-colinear points that are equidistant from 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.)
DimTarskiG Itv ..^ ..^

Definitiondf-trkg2d 23974* Define the class of geometries fulfilling the lower dimension axiom, Axiom A8 of [Schwabhauser] p. 12, and the upper dimension axiom, Axiom A9 of [Schwabhauser] p. 13, for dimension 2. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiG2D Itv

Definitiondf-trkg 23975* 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, where
• for betweenness, , where Itv
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2498.

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 (TarskiGBC)
• A5 5-segment axiom (TarskiGBC)
• A6 Identity for the betweenness relation (TarskiGB)
• A7 Axiom of Pasch (TarskiGB)
• A8 Lower dimension axiom (TarskiG2D)
• A9 Upper dimension axiom (TarskiG2D)
• 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 TarskiG2D
• 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 Itv LineG

Theoremistrkgc 23976* Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiGC

Theoremistrkgb 23977* Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiGB

Theoremistrkgcb 23978* Property of being a Tarski geometry - congruence and betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiGCB

Theoremistrkge 23979* Property of fulfilling Euclid's axiom (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiGE

Theoremistrkgl 23980* Building lines from the segment property (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       Itv LineG LineG

Theoremistrkg2d 23981* Property of fulfilling dimension 2 axiom (Contributed by Thierry Arnoux, 29-May-2019.)
Itv       TarskiG2D

Theoremistrkgld 23982* Property of fulfilling the lower dimension axiom (Contributed by Thierry Arnoux, 20-Nov-2019.)
Itv       DimTarskiG ..^ ..^

Theoremistrkg2ld 23983* Property of fulfilling the lower dimension 2 axiom (Contributed by Thierry Arnoux, 20-Nov-2019.)
Itv       DimTarskiG

Theoremaxtgcgrrflx 23984 Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiG

Theoremaxtgcgrid 23985 Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Itv       TarskiG

Theoremaxtgsegcon 23986* 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 , one can construct a line segment congruent to it, starting at any point and going in the direction of any ray containing . The ray is determined by the point and a second point , the endpoint of the ray. The other endpoint of the line segment to be constructed is just the point whose existence is asserted." (Contributed by Thierry Arnoux, 15-Mar-2019.)
Itv       TarskiG

Theoremaxtg5seg 23987 Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles and , a point on , and a point on . If all corresponding line segments except for and are congruent ( i.e., , , , and ), then and 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.)
Itv       TarskiG

Theoremaxtgbtwnid 23988 Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Itv       TarskiG

Theoremaxtgpasch 23989* Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle , point in segment , and point in segment , there exists a point on both the segment and the segment . 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.)
Itv       TarskiG

Theoremaxtgcont1 23990* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom (scheme) asserts that any two sets and (of points) such that the elements of precede the elements of with respect to some point (that is, is between and whenever is in and is in ) are separated by some point ; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Itv       TarskiG

Theoremaxtgcont 23991* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 23990. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Itv       TarskiG

Theoremaxtglowdim2OLD 23992* TODO-NM: What shall we do with such OLD theorems? Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points/ (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Itv       TarskiG2D

Theoremaxtglowdim2 23993* 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.)
Itv              DimTarskiG

Theoremaxtgupdim2 23994 Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 29-May-2019.)
Itv       TarskiG2D

Theoremaxtgeucl 23995* 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.)
Itv       TarskiGE

15.2  Tarskian Geometry

15.2.1  Congruence

Theoremtgcgrcomlr 23996 Congruence commutes on both sides. Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgcgreqb 23997 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgreq 23998 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgrneq 23999 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgrtriv 24000 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

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