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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | no2times 28401 | Version of 2times 12402 for surreal numbers. (Contributed by Scott Fenton, 23-Jul-2025.) |
| ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴)) | ||
| Theorem | 2nns 28402 | Surreal two is a surreal natural. (Contributed by Scott Fenton, 23-Jul-2025.) |
| ⊢ 2s ∈ ℕs | ||
| Theorem | 2sno 28403 | Surreal two is a surreal number. (Contributed by Scott Fenton, 23-Jul-2025.) |
| ⊢ 2s ∈ No | ||
| Theorem | 2ne0s 28404 | Surreal two is non-zero. (Contributed by Scott Fenton, 23-Jul-2025.) |
| ⊢ 2s ≠ 0s | ||
| Theorem | n0seo 28405* | 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 28406* | 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 28407 | 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 28408 | 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 28409 | 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 28410 | Surreal exponentiation to zero. (Contributed by Scott Fenton, 24-Jul-2025.) |
| ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s ) | ||
| Theorem | exps1 28411 | Surreal exponentiation to one. (Contributed by Scott Fenton, 24-Jul-2025.) |
| ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴) | ||
| Theorem | expsp1 28412 | 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 28413 | Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.) |
| ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No ) | ||
| Theorem | expsne0 28414 | 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 28415 | 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 28416 | 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 28417 | 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 28418 | 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 28419 | 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 28420 | Extend halfcut 28416 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 28421* | 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 28422 | The class of dyadic fractions is a set. (Contributed by Scott Fenton, 7-Aug-2025.) |
| ⊢ ℤs[1/2] ∈ V | ||
| Theorem | zzs12 28423 | A surreal integer is a dyadic fraction. (Contributed by Scott Fenton, 7-Aug-2025.) |
| ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2]) | ||
| Theorem | zs12bday 28424 | A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.) |
| ⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω) | ||
| Syntax | creno 28425 | Declare the syntax for the surreal reals. |
| class ℝs | ||
| Definition | df-reno 28426* | 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 28427* | 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 28428* | 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 28429 | Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.) |
| ⊢ 0s ∈ ℝs | ||
| Theorem | renegscl 28430 | 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 28431 | 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 28432* | Lemma for remulscl 28434. 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 28433* | Lemma for remulscl 28434. 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 28434 | 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 28461 et sequentes). This allows to treat a Tarski structure as a special kind of extensible structure (see df-struct 17184). 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 28483 and tgjustc2 28484, which in turn are supported by tgjustf 28481 and tgjustr 28482. 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/ 28482. 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 1547 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 28461), of "Euclidean dimensionless Tarski structures" (df-trkge 28459) and of "Tarski structures of dimension no less than N" (df-trkgld 28460). In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 28816 and iscgra 28817). 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 28519. 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 28435 | Extends class notation with the class of Tarski geometries. |
| class TarskiG | ||
| Syntax | cstrkgc 28436 | Extends class notation with the class of geometries fulfilling the congruence axioms. |
| class TarskiGC | ||
| Syntax | cstrkgb 28437 | Extends class notation with the class of geometries fulfilling the betweenness axioms. |
| class TarskiGB | ||
| Syntax | cstrkgcb 28438 | Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms. |
| class TarskiGCB | ||
| Syntax | cstrkgld 28439 | Extends class notation with the relation for geometries fulfilling the lower dimension axioms. |
| class DimTarskiG≥ | ||
| Syntax | cstrkge 28440 | Extends class notation with the class of geometries fulfilling Euclid's axiom. |
| class TarskiGE | ||
| Syntax | citv 28441 | Declare the syntax for the Interval (segment) index extractor. |
| class Itv | ||
| Syntax | clng 28442 | Declare the syntax for the Line function. |
| class LineG | ||
| Definition | df-itv 28443 | Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent form itvid 28447 instead. (New usage is discouraged.) |
| ⊢ Itv = Slot ;16 | ||
| Definition | df-lng 28444 | Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.) Use its index-independent form lngid 28448 instead. (New usage is discouraged.) |
| ⊢ LineG = Slot ;17 | ||
| Theorem | itvndx 28445 | Index value of the Interval (segment) slot. Use ndxarg 17233. (Contributed by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.) |
| ⊢ (Itv‘ndx) = ;16 | ||
| Theorem | lngndx 28446 | Index value of the "line" slot. Use ndxarg 17233. (Contributed by Thierry Arnoux, 27-Mar-2019.) (New usage is discouraged.) |
| ⊢ (LineG‘ndx) = ;17 | ||
| Theorem | itvid 28447 | Utility theorem: index-independent form of df-itv 28443. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
| ⊢ Itv = Slot (Itv‘ndx) | ||
| Theorem | lngid 28448 | Utility theorem: index-independent form of df-lng 28444. (Contributed by Thierry Arnoux, 27-Mar-2019.) |
| ⊢ LineG = Slot (LineG‘ndx) | ||
| Theorem | slotsinbpsd 28449 | The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 28885 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 28450 | The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 28885 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 28451 | The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 28883. (Contributed by AV, 9-Nov-2024.) |
| ⊢ (LineG‘ndx) ≠ (Itv‘ndx) | ||
| Theorem | trkgstr 28452 | 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 28453 | 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 28454 | 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 28455 | 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 28456* | 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 2760, 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 28457* | 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 28458* | 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 28459* | 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 28460* | 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 28461* |
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 28462* | 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 28463* | 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 28464* | 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 28465* | 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 28466* | 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 28467* | 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 28468* | 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 28469* | 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 28470 | 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 28471 | 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 28472* | 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 28473 | 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 28474 | 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 28475* | 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 28476* | 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 28477* | Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 28476. (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 28478* | 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 28479 | 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 28480* | 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 28481* | 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 28482* | 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 28483* | 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 28484* | 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 28485 | 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 28486 | 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)) | ||
| Theorem | tgcgrcoml 28487 | 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 = (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 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) | ||
| Theorem | tgcgrcomlr 28488 | Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
| ⊢ 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 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) | ||
| Theorem | tgcgreqb 28489 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
| ⊢ 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 ↔ 𝐶 = 𝐷)) | ||
| Theorem | tgcgreq 28490 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
| ⊢ 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) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
| Theorem | tgcgrneq 28491 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
| ⊢ 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) ⇒ ⊢ (𝜑 → 𝐶 ≠ 𝐷) | ||
| Theorem | tgcgrtriv 28492 | Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵)) | ||
| Theorem | tgcgrextend 28493 | 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 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) | ||
| Theorem | tgsegconeq 28494 | Two points that satisfy the conclusion of axtgsegcon 28472 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ≠ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐸)) & ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐴 − 𝐹) = (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐸 = 𝐹) | ||
| Theorem | tgbtwntriv2 28495 | Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) | ||
| Theorem | tgbtwncom 28496 | Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) | ||
| Theorem | tgbtwncomb 28497 | Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) | ||
| Theorem | tgbtwnne 28498 | Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
| Theorem | tgbtwntriv1 28499 | Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) | ||
| Theorem | tgbtwnswapid 28500 | If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.) |
| ⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
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