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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elnnzs 28401 | Positive surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁)) | ||
Theorem | elznns 28402 | Surreal integer property expressed in terms of positive integers and non-negative integers. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕ0s))) | ||
Theorem | zn0subs 28403 | The non-negative difference of surreal integers is a non-negative integer. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s)) | ||
Theorem | peano5uzs 28404* | Peano's inductive postulate for upper surreal integers. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ (𝜑 → 𝑁 ∈ ℤs) & ⊢ (𝜑 → 𝑁 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴) ⇒ ⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴) | ||
Theorem | uzsind 28405* | Induction on the upper surreal integers that start at 𝑀. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 +s 1s ) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤs → 𝜓) & ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏) | ||
Theorem | zsbday 28406 | A surreal integer has a finite birthday. (Contributed by Scott Fenton, 26-May-2025.) |
⊢ (𝐴 ∈ ℤs → ( bday ‘𝐴) ∈ ω) | ||
Theorem | zscut 28407 | A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.) |
⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) | ||
Syntax | c2s 28408 | Declare the syntax for surreal two. |
class 2s | ||
Definition | df-2s 28409 | Define surreal two. This is the simplest number greater than one. See 1p1e2s 28414 for its addition version. (Contributed by Scott Fenton, 27-May-2025.) |
⊢ 2s = ({ 1s } |s ∅) | ||
Syntax | cexps 28410 | Declare the syntax for surreal exponentiation. |
class ↑s | ||
Definition | df-exps 28411* | Define surreal exponentiation. Compare df-exp 14099. (Contributed by Scott Fenton, 27-May-2025.) |
⊢ ↑s = (𝑥 ∈ No , 𝑦 ∈ ℤs ↦ if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)))))) | ||
Syntax | czs12 28412 | Define the syntax for the set of surreal dyadic fractions. |
class ℤs[1/2] | ||
Definition | df-zs12 28413* | Define the set of dyadic rationals. This is the set of rationals whose denominator is a power of two. Later we will prove that this is precisely the set of surreals with a finite birthday. (Contributed by Scott Fenton, 27-May-2025.) |
⊢ ℤs[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs ∃𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2s↑s𝑧))} | ||
Theorem | 1p1e2s 28414 | One plus one is two. Surreal version. (Contributed by Scott Fenton, 27-May-2025.) |
⊢ ( 1s +s 1s ) = 2s | ||
Theorem | no2times 28415 | Version of 2times 12399 for surreal numbers. (Contributed by Scott Fenton, 23-Jul-2025.) |
⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴)) | ||
Theorem | 2nns 28416 | Surreal two is a surreal natural. (Contributed by Scott Fenton, 23-Jul-2025.) |
⊢ 2s ∈ ℕs | ||
Theorem | 2sno 28417 | Surreal two is a surreal number. (Contributed by Scott Fenton, 23-Jul-2025.) |
⊢ 2s ∈ No | ||
Theorem | 2ne0s 28418 | Surreal two is non-zero. (Contributed by Scott Fenton, 23-Jul-2025.) |
⊢ 2s ≠ 0s | ||
Theorem | n0seo 28419* | A non-negative surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.) |
⊢ (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s ))) | ||
Theorem | zseo 28420* | A surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.) |
⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s ))) | ||
Theorem | nohalf 28421 | An explicit expression for one half. This theorem avoids the axiom of infinity. (Contributed by Scott Fenton, 23-Jul-2025.) |
⊢ ( 1s /su 2s) = ({ 0s } |s { 1s }) | ||
Theorem | expsval 28422 | The value of surreal exponentiation. (Contributed by Scott Fenton, 24-Jul-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ℤs) → (𝐴↑s𝐵) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))))) | ||
Theorem | expsnnval 28423 | Value of surreal exponentiation at a natural number. (Contributed by Scott Fenton, 25-Jul-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁)) | ||
Theorem | exps0 28424 | Surreal exponentiation to zero. (Contributed by Scott Fenton, 24-Jul-2025.) |
⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s ) | ||
Theorem | exps1 28425 | Surreal exponentiation to one. (Contributed by Scott Fenton, 24-Jul-2025.) |
⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴) | ||
Theorem | expsp1 28426 | Value of a surreal number raised to a non-negative integer power plus one. (Contributed by Scott Fenton, 6-Aug-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴)) | ||
Theorem | expscl 28427 | Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No ) | ||
Theorem | expsne0 28428 | A non-negative surreal integer power is non-zero if its base is non-zero. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s ) | ||
Theorem | expsgt0 28429 | A non-negative surreal integer power is positive if its base is positive. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁)) | ||
Theorem | halfcut 28430 | Relate the cut of twice of two numbers to the cut of the numbers. Lemma 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵)) & ⊢ 𝐶 = ({𝐴} |s {𝐵}) ⇒ ⊢ (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s)) | ||
Theorem | cutpw2 28431 | A cut expression for inverses of powers of two. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ (𝑁 ∈ ℕ0s → ( 1s /su (2s↑s(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑁))})) | ||
Theorem | pw2bday 28432 | The inverses of powers of two have finite birthdays. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ (𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω) | ||
Theorem | addhalfcut 28433 | The cut of a surreal non-negative integer and its successor is the original number plus one half. Part of theorem 4.2 of [Gonshor] p. 30. (Contributed by Scott Fenton, 13-Aug-2025.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0s) ⇒ ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s))) | ||
Theorem | pw2cut 28434 | Extend halfcut 28430 to arbitrary powers of two. Part of theorem 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 18-Aug-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0s) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵)) ⇒ ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s )))) | ||
Theorem | elzs12 28435* | Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))) | ||
Theorem | zs12ex 28436 | The class of dyadic fractions is a set. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ ℤs[1/2] ∈ V | ||
Theorem | zzs12 28437 | A surreal integer is a dyadic fraction. (Contributed by Scott Fenton, 7-Aug-2025.) |
⊢ (𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2]) | ||
Theorem | zs12bday 28438 | A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.) |
⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω) | ||
Syntax | creno 28439 | Declare the syntax for the surreal reals. |
class ℝs | ||
Definition | df-reno 28440* | Define the surreal reals. These are the finite numbers without any infintesimal parts. Definition from [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ ℝs = {𝑥 ∈ No ∣ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑥 ∧ 𝑥 <s 𝑛) ∧ 𝑥 = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (𝑥 -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (𝑥 +s ( 1s /su 𝑛))}))} | ||
Theorem | elreno 28441* | Membership in the set of surreal reals. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})))) | ||
Theorem | recut 28442* | The cut involved in defining surreal reals is a genuine cut. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) | ||
Theorem | 0reno 28443 | Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ 0s ∈ ℝs | ||
Theorem | renegscl 28444 | The surreal reals are closed under negation. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ (𝐴 ∈ ℝs → ( -us ‘𝐴) ∈ ℝs) | ||
Theorem | readdscl 28445 | The surreal reals are closed under addition. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.) |
⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 +s 𝐵) ∈ ℝs) | ||
Theorem | remulscllem1 28446* | Lemma for remulscl 28448. Split a product of reciprocals of naturals. (Contributed by Scott Fenton, 16-Apr-2025.) |
⊢ (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛))) | ||
Theorem | remulscllem2 28447* | Lemma for remulscl 28448. Bound 𝐴 and 𝐵 above and below. (Contributed by Scott Fenton, 16-Apr-2025.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ((𝑁 ∈ ℕs ∧ 𝑀 ∈ ℕs) ∧ ((( -us ‘𝑁) <s 𝐴 ∧ 𝐴 <s 𝑁) ∧ (( -us ‘𝑀) <s 𝐵 ∧ 𝐵 <s 𝑀)))) → ∃𝑝 ∈ ℕs (( -us ‘𝑝) <s (𝐴 ·s 𝐵) ∧ (𝐴 ·s 𝐵) <s 𝑝)) | ||
Theorem | remulscl 28448 | The surreal reals are closed under multiplication. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 16-Apr-2025.) |
⊢ ((𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs) → (𝐴 ·s 𝐵) ∈ ℝs) | ||
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) (Base‘𝑓) together with functions (Itv‘𝑓) and (dist‘𝑓) on ((Base‘𝑓) × (Base‘𝑓)) satisfying certain axioms (given in Definitions df-trkg 28475 et sequentes). This allows to treat a Tarski structure as a special kind of extensible structure (see df-struct 17180). 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 𝑥, 𝑦, 𝑧 ∈ (Base‘𝑓), one has "Btw 𝑥𝑦𝑧 " if and only if 𝑦 ∈ 𝑥(Itv‘𝑓)𝑧, and a congruence quaternary predicate Congr by positing that, for any 𝑥, 𝑦, 𝑧, 𝑡 ∈ (Base‘𝑓), one has "Congr 𝑥𝑦𝑧𝑡 " if and only if 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡. It is easy to check that if 𝑓 satisfies our Tarski axioms, then Btw and Congr satisfy Tarski's Tarski axioms when (Base‘𝑓) is interpreted as the universe of discourse. Conversely, suppose that one is given a set 𝑎, a ternary predicate Btw, and a quaternary predicate Congr. One defines the extensible structure 𝑓 such that (Base‘𝑓) is 𝑎, and (Itv‘𝑓) is the function which associates with each 〈𝑥, 𝑦〉 ∈ (𝑎 × 𝑎) the set of points 𝑧 ∈ 𝑎 such that "Btw 𝑥𝑧𝑦", and (dist‘𝑓) is the function which associates with each 〈𝑥, 𝑦〉 ∈ (𝑎 × 𝑎) the set of ordered pairs 〈𝑧, 𝑡〉 ∈ (𝑎 × 𝑎) such that "Congr 𝑥𝑦𝑧𝑡". It is easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when 𝑎 is interpreted as the universe of discourse, then 𝑓 satisfies our Tarski axioms. We intentionally choose to represent congruence (without loss of generality) as 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 instead of "Congr 𝑥𝑦𝑧𝑡", as it is more convenient. It is always possible to define dist for any particular geometry to produce equal results when congruence 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 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 causes no loss of generality in tgjustc1 28497 and tgjustc2 28498, which in turn are supported by tgjustf 28495 and tgjustr 28496. 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/ 28496. 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 𝜑 →) by insert a ⊤ → in each hypothesis, using a1i 11, then using mptru 1543 to remove the final ⊤ → 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 28475), of "Euclidean dimensionless Tarski structures" (df-trkge 28473) and of "Tarski structures of dimension no less than N" (df-trkgld 28474). In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 28830 and iscgra 28831). 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 28533. 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. | ||
Syntax | cstrkg 28449 | Extends class notation with the class of Tarski geometries. |
class TarskiG | ||
Syntax | cstrkgc 28450 | Extends class notation with the class of geometries fulfilling the congruence axioms. |
class TarskiGC | ||
Syntax | cstrkgb 28451 | Extends class notation with the class of geometries fulfilling the betweenness axioms. |
class TarskiGB | ||
Syntax | cstrkgcb 28452 | Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms. |
class TarskiGCB | ||
Syntax | cstrkgld 28453 | Extends class notation with the relation for geometries fulfilling the lower dimension axioms. |
class DimTarskiG≥ | ||
Syntax | cstrkge 28454 | Extends class notation with the class of geometries fulfilling Euclid's axiom. |
class TarskiGE | ||
Syntax | citv 28455 | Declare the syntax for the Interval (segment) index extractor. |
class Itv | ||
Syntax | clng 28456 | Declare the syntax for the Line function. |
class LineG | ||
Definition | df-itv 28457 | Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent form itvid 28461 instead. (New usage is discouraged.) |
⊢ Itv = Slot ;16 | ||
Definition | df-lng 28458 | Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.) Use its index-independent form lngid 28462 instead. (New usage is discouraged.) |
⊢ LineG = Slot ;17 | ||
Theorem | itvndx 28459 | Index value of the Interval (segment) slot. Use ndxarg 17229. (Contributed by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.) |
⊢ (Itv‘ndx) = ;16 | ||
Theorem | lngndx 28460 | Index value of the "line" slot. Use ndxarg 17229. (Contributed by Thierry Arnoux, 27-Mar-2019.) (New usage is discouraged.) |
⊢ (LineG‘ndx) = ;17 | ||
Theorem | itvid 28461 | Utility theorem: index-independent form of df-itv 28457. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ Itv = Slot (Itv‘ndx) | ||
Theorem | lngid 28462 | Utility theorem: index-independent form of df-lng 28458. (Contributed by Thierry Arnoux, 27-Mar-2019.) |
⊢ LineG = Slot (LineG‘ndx) | ||
Theorem | slotsinbpsd 28463 | The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 28899 and proofs using it. (Contributed by AV, 29-Oct-2024.) |
⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) | ||
Theorem | slotslnbpsd 28464 | The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 28899 and proofs using it. (Contributed by AV, 29-Oct-2024.) |
⊢ (((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) | ||
Theorem | lngndxnitvndx 28465 | The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 28897. (Contributed by AV, 9-Nov-2024.) |
⊢ (LineG‘ndx) ≠ (Itv‘ndx) | ||
Theorem | trkgstr 28466 | Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} ⇒ ⊢ 𝑊 Struct 〈1, ;16〉 | ||
Theorem | trkgbas 28467 | The base set of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝑊)) | ||
Theorem | trkgdist 28468 | The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) | ||
Theorem | trkgitv 28469 | The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (Itv‘𝑊)) | ||
Definition | df-trkgc 28470* | 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 2757, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
⊢ TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))} | ||
Definition | df-trkgb 28471* | 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 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))} | ||
Definition | df-trkgcb 28472* | 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 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∀𝑐 ∈ 𝑝 ∀𝑣 ∈ 𝑝 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))} | ||
Definition | df-trkge 28473* | Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ TarskiGE = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))} | ||
Definition | df-trkgld 28474* | 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 − 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.) |
⊢ DimTarskiG≥ = {〈𝑔, 𝑛〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑓(𝑓:(1..^𝑛)–1-1→𝑝 ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓‘𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓‘𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓‘𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))} | ||
Definition | df-trkg 28475* |
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:
Tarski originally had more axioms, but later reduced his list to 11:
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 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) | ||
Theorem | istrkgc 28476* | Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))) | ||
Theorem | istrkgb 28477* | Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦))))) | ||
Theorem | istrkgcb 28478* | Property of being a Tarski geometry - congruence and betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∀𝑐 ∈ 𝑃 ∀𝑣 ∈ 𝑃 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 − 𝑦) = (𝑎 − 𝑏) ∧ (𝑦 − 𝑧) = (𝑏 − 𝑐)) ∧ ((𝑥 − 𝑢) = (𝑎 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑏 − 𝑣)))) → (𝑧 − 𝑢) = (𝑐 − 𝑣)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))) | ||
Theorem | istrkge 28479* | Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiGE ↔ (𝐺 ∈ V ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏))))) | ||
Theorem | istrkgl 28480* | Building lines from the segment property. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))) | ||
Theorem | istrkgld 28481* | Property of fulfilling the lower dimension 𝑁 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝐺DimTarskiG≥𝑁 ↔ ∃𝑓(𝑓:(1..^𝑁)–1-1→𝑃 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) − 𝑥) = ((𝑓‘𝑗) − 𝑥) ∧ ((𝑓‘1) − 𝑦) = ((𝑓‘𝑗) − 𝑦) ∧ ((𝑓‘1) − 𝑧) = ((𝑓‘𝑗) − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))) | ||
Theorem | istrkg2ld 28482* | Property of fulfilling the lower dimension 2 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐺DimTarskiG≥2 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) | ||
Theorem | istrkg3ld 28483* | Property of fulfilling the lower dimension 3 axiom. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐺DimTarskiG≥3 ↔ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))) | ||
Theorem | axtgcgrrflx 28484 | Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑌 − 𝑋)) | ||
Theorem | axtgcgrid 28485 | Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | axtgsegcon 28486* | 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 𝐴𝐵, one can construct a line segment congruent to it, starting at any point 𝑌 and going in the direction of any ray containing 𝑌. The ray is determined by the point 𝑌 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))) | ||
Theorem | axtg5seg 28487 | Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles 𝑋𝑍𝑈 and 𝐴𝐶𝑉, a point 𝑌 on 𝑋𝑍, and a point 𝐵 on 𝐴𝐶. If all corresponding line segments except for 𝑍𝑈 and 𝐶𝑉 are congruent ( i.e., 𝑋𝑌 ∼ 𝐴𝐵, 𝑌𝑍 ∼ 𝐵𝐶, 𝑋𝑈 ∼ 𝐴𝑉, and 𝑌𝑈 ∼ 𝐵𝑉), then 𝑍𝑈 and 𝐶𝑉 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐴 − 𝐵)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝑋 − 𝑈) = (𝐴 − 𝑉)) & ⊢ (𝜑 → (𝑌 − 𝑈) = (𝐵 − 𝑉)) ⇒ ⊢ (𝜑 → (𝑍 − 𝑈) = (𝐶 − 𝑉)) | ||
Theorem | axtgbtwnid 28488 | Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | axtgpasch 28489* | Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle 𝑋𝑌𝑍, point 𝑈 in segment 𝑋𝑍, and point 𝑉 in segment 𝑌𝑍, there exists a point 𝑎 on both the segment 𝑈𝑌 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ (𝑋𝐼𝑍)) & ⊢ (𝜑 → 𝑉 ∈ (𝑌𝐼𝑍)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋))) | ||
Theorem | axtgcont1 28490* | 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 𝑎 (that is, 𝑥 is between 𝑎 and 𝑦 whenever 𝑥 is in 𝑋 and 𝑦 is in 𝑌) are separated by some point 𝑏; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑆 ⊆ 𝑃) & ⊢ (𝜑 → 𝑇 ⊆ 𝑃) ⇒ ⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))) | ||
Theorem | axtgcont 28491* | Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 28490. (Contributed by Thierry Arnoux, 16-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑆 ⊆ 𝑃) & ⊢ (𝜑 → 𝑇 ⊆ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) → 𝑢 ∈ (𝐴𝐼𝑣)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)) | ||
Theorem | axtglowdim2 28492* | 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) | ||
Theorem | axtgupdim2 28493 | Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. Three points 𝑋, 𝑌 and 𝑍 equidistant to two given two points 𝑈 and 𝑉 must be colinear. (Contributed by Thierry Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑈 − 𝑋) = (𝑉 − 𝑋)) & ⊢ (𝜑 → (𝑈 − 𝑌) = (𝑉 − 𝑌)) & ⊢ (𝜑 → (𝑈 − 𝑍) = (𝑉 − 𝑍)) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥3) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) | ||
Theorem | axtgeucl 28494* | 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiGE) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ (𝑋𝐼𝑉)) & ⊢ (𝜑 → 𝑈 ∈ (𝑌𝐼𝑍)) & ⊢ (𝜑 → 𝑋 ≠ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑎) ∧ 𝑍 ∈ (𝑋𝐼𝑏) ∧ 𝑉 ∈ (𝑎𝐼𝑏))) | ||
Theorem | tgjustf 28495* | Given any function 𝐹, equality of the image by 𝐹 is an equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))) | ||
Theorem | tgjustr 28496* | 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) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦)))) | ||
Theorem | tgjustc1 28497* | 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) ⇒ ⊢ ∃𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (〈𝑤, 𝑥〉𝑟〈𝑦, 𝑧〉 ↔ (𝑤 − 𝑥) = (𝑦 − 𝑧))) | ||
Theorem | tgjustc2 28498* | 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 = (Base‘𝐺) & ⊢ 𝑅 Er (𝑃 × 𝑃) ⇒ ⊢ ∃𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤 ∈ 𝑃 ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 (〈𝑤, 𝑥〉𝑅〈𝑦, 𝑧〉 ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧))) | ||
Theorem | tgcgrcomimp 28499 | Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by David A. Wheeler, 29-Jun-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶))) | ||
Theorem | tgcgrcomr 28500 | 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶)) |
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