P. 288 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	exps1 28701	Surreal exponentiation to one. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
Theorem	expsp1 28702	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	expscllem 28703*	Lemma for proving non-negative surreal integer exponentiation closure. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ 𝐹 ⊆ No     &   ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ·s 𝑦) ∈ 𝐹)    &   ⊢ 1s ∈ 𝐹    ⇒   ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ 𝐹)
Theorem	expscl 28704	Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No )
Theorem	n0expscl 28705	Closure law for non-negative surreal integer exponentiation. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕ0s)
Theorem	nnexpscl 28706	Closure law for positive surreal integer exponentiation. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ((𝐴 ∈ ℕs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕs)
Theorem	zexpscl 28707	Closure law for surreal integer exponentiation. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℤs)
Theorem	expadds 28708	Sum of exponents law for surreals. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
Theorem	expsne0 28709	A non-negative surreal integer power is nonzero if its base is nonzero. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )
Theorem	expsgt0 28710	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	pw2recs 28711*	Any power of two has a multiplicative inverse. Note that this theorem does not require the axiom of infinity. (Contributed by Scott Fenton, 5-Sep-2025.)
⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )
Theorem	pw2divscld 28712	Division closure for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (𝐴 /su (2s↑s𝑁)) ∈ No )
Theorem	pw2divmulsd 28713	Relationship between surreal division and multiplication for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((𝐴 /su (2s↑s𝑁)) = 𝐵 ↔ ((2s↑s𝑁) ·s 𝐵) = 𝐴))
Theorem	pw2divscan3d 28714	Cancellation law for surreal division by powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (((2s↑s𝑁) ·s 𝐴) /su (2s↑s𝑁)) = 𝐴)
Theorem	pw2divscan2d 28715	A cancellation law for surreal division by powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((2s↑s𝑁) ·s (𝐴 /su (2s↑s𝑁))) = 𝐴)
Theorem	pw2divsassd 28716	An associative law for division by powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) /su (2s↑s𝑁)) = (𝐴 ·s (𝐵 /su (2s↑s𝑁))))
Theorem	pw2divscan4d 28717	Cancellation law for divison by powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (𝐴 /su (2s↑s𝑁)) = (((2s↑s𝑀) ·s 𝐴) /su (2s↑s(𝑁 +s 𝑀))))
Theorem	pw2gt0divsd 28718	Division of a positive surreal by a power of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ( 0s <s 𝐴 ↔ 0s <s (𝐴 /su (2s↑s𝑁))))
Theorem	pw2ge0divsd 28719	Divison of a non-negative surreal by a power of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ( 0s ≤s 𝐴 ↔ 0s ≤s (𝐴 /su (2s↑s𝑁))))
Theorem	pw2divsrecd 28720	Relationship between surreal division and reciprocal for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (𝐴 /su (2s↑s𝑁)) = (𝐴 ·s ( 1s /su (2s↑s𝑁))))
Theorem	pw2divsdird 28721	Distribution of surreal division over addition for powers of two. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) /su (2s↑s𝑁)) = ((𝐴 /su (2s↑s𝑁)) +s (𝐵 /su (2s↑s𝑁))))
Theorem	pw2divsnegd 28722	Move negative sign inside of a power of two division. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ( -us ‘(𝐴 /su (2s↑s𝑁))) = (( -us ‘𝐴) /su (2s↑s𝑁)))
Theorem	pw2ltdivmulsd 28723	Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((𝐴 /su (2s↑s𝑁)) <s 𝐵 ↔ 𝐴 <s ((2s↑s𝑁) ·s 𝐵)))
Theorem	pw2ltmuldivs2d 28724	Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (((2s↑s𝑁) ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su (2s↑s𝑁))))
Theorem	pw2ltsdiv1d 28725	Surreal less-than relationship for division by a power of two. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 /su (2s↑s𝑁)) <s (𝐵 /su (2s↑s𝑁))))
Theorem	avglts1d 28726	Ordering property for average. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ 𝐴 <s ((𝐴 +s 𝐵) /su 2s)))
Theorem	avglts2d 28727	Ordering property for average. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ((𝐴 +s 𝐵) /su 2s) <s 𝐵))
Theorem	pw2divs0d 28728	Division into zero is zero for a power of two. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ( 0s /su (2s↑s𝑁)) = 0s )
Theorem	pw2divsidd 28729	Identity law for division over powers of two. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((2s↑s𝑁) /su (2s↑s𝑁)) = 1s )
Theorem	pw2ltdivmuls2d 28730	Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ((𝐴 /su (2s↑s𝑁)) <s 𝐵 ↔ 𝐴 <s (𝐵 ·s (2s↑s𝑁))))
Theorem	halfcut 28731	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.) Avoid the axiom of infinity. (Proof modified by Scott Fenton, 6-Sep-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	addhalfcut 28732	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 28733	Extend halfcut 28731 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	pw2cutp1 28734	Simplify pw2cut 28733 in the case of successors of surreal integers. (Contributed by Scott Fenton, 11-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))))
Theorem	pw2cut2 28735	Cut expression for powers of two. Theorem 12 of [Conway] p. 12-13. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))
Theorem	bdaypw2n0bndlem 28736	Lemma for bdaypw2n0bnd 28737. Prove the case with a successor. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s(𝑁 +s 1s ))) → ( bday ‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc ( bday ‘(𝑁 +s 1s )))
Theorem	bdaypw2n0bnd 28737	Upper bound for the birthday of a proper fraction of a power of two. This is actually a strict equality when 𝐴 is odd, but we do not need this for the rest of our development. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s𝑁)) → ( bday ‘(𝐴 /su (2s↑s𝑁))) ⊆ suc ( bday ‘𝑁))
Theorem	bdaypw2bnd 28738	Birthday bounding rule for non-negative dyadic rationals. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑋 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑌 <s (2s↑s𝑃))    &   ⊢ (𝜑 → (𝑋 +s 𝑃) <s 𝑁)    ⇒   ⊢ (𝜑 → ( bday ‘(𝑋 +s (𝑌 /su (2s↑s𝑃)))) ⊆ ( bday ‘𝑁))
Theorem	bdayfinbndcbv 28739*	Lemma for bdayfinbnd 28742. Change some bound variables. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))    ⇒   ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s 𝑁))))
Theorem	bdayfinbndlem1 28740*	Lemma for bdayfinbnd 28742. Show the first half of the inductive step. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))    ⇒   ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
Theorem	bdayfinbndlem2 28741*	Lemma for bdayfinbnd 28742. Conduct the induction. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
Theorem	bdayfinbnd 28742*	Given a non-negative integer and a non-negative surreal of lesser or equal birthday, show that the surreal can be expressed as a dyadic fraction with an upper bound on the integer and exponent. This proof follows the proof from Mizar at https://mizar.uwb.edu.pl/version/current/html/surrealn.html. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → ( bday ‘𝑍) ⊆ ( bday ‘𝑁))    &   ⊢ (𝜑 → 0s ≤s 𝑍)    ⇒   ⊢ (𝜑 → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
Theorem	z12bdaylem1 28743	Lemma for z12bday 28758. Prove an inequality for birthday ordering. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0s)    &   ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))    ⇒   ⊢ (𝜑 → (𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃))) ≠ (𝑁 +s 𝑃))
Theorem	z12bdaylem2 28744	Lemma for z12bday 28758. Show the first half of the equality. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0s)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0s)    &   ⊢ (𝜑 → ((2s ·s 𝑀) +s 1s ) <s (2s↑s𝑃))    ⇒   ⊢ (𝜑 → ( bday ‘(𝑁 +s (((2s ·s 𝑀) +s 1s ) /su (2s↑s𝑃)))) ⊆ ( bday ‘((𝑁 +s 𝑃) +s 1s )))
Theorem	elz12s 28745*	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	elz12si 28746	Inference form of membership in the dyadic fractions. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2])
Theorem	z12sex 28747	The class of dyadic fractions is a set. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ℤs[1/2] ∈ V
Theorem	zz12s 28748	A surreal integer is a dyadic fraction. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ (𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2])
Theorem	z12no 28749	A dyadic is a surreal. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ No )
Theorem	z12addscl 28750	The dyadics are closed under addition. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ ((𝐴 ∈ ℤs[1/2] ∧ 𝐵 ∈ ℤs[1/2]) → (𝐴 +s 𝐵) ∈ ℤs[1/2])
Theorem	z12negscl 28751	The dyadics are closed under negation. (Contributed by Scott Fenton, 9-Nov-2025.)
⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])
Theorem	z12subscl 28752	The dyadics are closed under subtraction. (Contributed by Scott Fenton, 12-Dec-2025.)
⊢ ((𝐴 ∈ ℤs[1/2] ∧ 𝐵 ∈ ℤs[1/2]) → (𝐴 -s 𝐵) ∈ ℤs[1/2])
Theorem	z12shalf 28753	Half of a dyadic is a dyadic. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])
Theorem	z12negsclb 28754	A surreal is a dyadic fraction iff its negative is. (Contributed by Scott Fenton, 9-Nov-2025.)
⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( -us ‘𝐴) ∈ ℤs[1/2]))
Theorem	z12zsodd 28755*	A dyadic fraction is either an integer or an odd number divided by a positive power of two. (Contributed by Scott Fenton, 5-Dec-2025.)
⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
Theorem	z12sge0 28756*	An expression for non-negative dyadic rationals. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
Theorem	z12bdaylem 28757	Lemma for z12bday 28758. Handle the non-negative case. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ ℤs[1/2] ∧ 0s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)
Theorem	z12bday 28758	A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.) (Proof shortened by Scott Fenton, 22-Feb-2026.)
⊢ (𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
Theorem	bdayfinlem 28759	Lemma for bdayfin 28760. Handle the non-negative case. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])
Theorem	bdayfin 28760	A surreal has a finite birthday iff it is a dyadic fraction. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝐴 ∈ No → (𝐴 ∈ ℤs[1/2] ↔ ( bday ‘𝐴) ∈ ω))
Theorem	dfz12s2 28761	The set of dyadic fractions is the same as the old set of ω. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ ℤs[1/2] = ( O ‘ω)
15.6.6  Real numbers
Syntax	creno 28762	Declare the syntax for the surreal reals.
class ℝs
Definition	df-reno 28763*	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 28764*	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	reno 28765	A surreal real is a surreal number. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )
Theorem	renod 28766	A surreal real is a surreal number. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	recut 28767*	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	elreno2 28768*	Alternate characterization of the surreal reals. Theorem 4.4(b) of [Gonshor] p. 39. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)))))
Theorem	0reno 28769	Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 0s ∈ ℝs
Theorem	1reno 28770	Surreal one is a surreal real. (Contributed by Scott Fenton, 18-Feb-2026.)
⊢ 1s ∈ ℝs
Theorem	renegscl 28771	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 28772	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 28773*	Lemma for remulscl 28775. 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 28774*	Lemma for remulscl 28775. 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 28775	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)
PART 16  ELEMENTARY GEOMETRY This part develops elementary geometry based on Tarski's axioms, following [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the "points". It has two primitive notions, the ternary predicate of "betweenness" and the quaternary predicate of "congruence". To adapt this theory to the framework of set.mm, and to be able to talk of *a* Tarski structure as a space satisfying the given axioms, we use the following definition, stated informally: A Tarski structure 𝑓 is a set (of points) (Base‘𝑓) together with functions (Itv‘𝑓) and (dist‘𝑓) on ((Base‘𝑓) × (Base‘𝑓)) satisfying certain axioms (given in Definitions df-trkg 28802 et sequentes). This allows to treat a Tarski structure as a special kind of extensible structure (see df-struct 17245). 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 28824 and tgjustc2 28825, which in turn are supported by tgjustf 28822 and tgjustr 28823. 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/ 28823. 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 1577 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 490, the theorem as stated, and ex 418. 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 28802), of "Euclidean dimensionless Tarski structures" (df-trkge 28800) and of "Tarski structures of dimension no less than N" (df-trkgld 28801). In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 29202 and iscgra 29203). 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 28861. The first section is devoted to the definitions of these various structures. The second section ("Tarskian geometry") develops the synthetic treatment of geometry. The remaining sections prove that real Euclidean spaces and complex Hilbert spaces, with intended interpretations, are Euclidean Tarski structures. Most of the work in this part is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. See also the credits in the comment of each statement.
16.1  Definition and Tarski's Axioms of Geometry
Syntax	cstrkg 28776	Extends class notation with the class of Tarski geometries.
class TarskiG
Syntax	cstrkgc 28777	Extends class notation with the class of geometries fulfilling the congruence axioms.
class TarskiGC
Syntax	cstrkgb 28778	Extends class notation with the class of geometries fulfilling the betweenness axioms.
class TarskiGB
Syntax	cstrkgcb 28779	Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms.
class TarskiGCB
Syntax	cstrkgld 28780	Extends class notation with the relation for geometries fulfilling the lower dimension axioms.
class DimTarskiG≥
Syntax	cstrkge 28781	Extends class notation with the class of geometries fulfilling Euclid's axiom.
class TarskiGE
Syntax	citv 28782	Declare the syntax for the Interval (segment) index extractor.
class Itv
Syntax	clng 28783	Declare the syntax for the Line function.
class LineG
Definition	df-itv 28784	Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent form itvid 28788 instead. (New usage is discouraged.)
⊢ Itv = Slot ;16
Definition	df-lng 28785	Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.) Use its index-independent form lngid 28789 instead. (New usage is discouraged.)
⊢ LineG = Slot ;17
Theorem	itvndx 28786	Index value of the Interval (segment) slot. Use ndxarg 17294. (Contributed by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.)
⊢ (Itv‘ndx) = ;16
Theorem	lngndx 28787	Index value of the "line" slot. Use ndxarg 17294. (Contributed by Thierry Arnoux, 27-Mar-2019.) (New usage is discouraged.)
⊢ (LineG‘ndx) = ;17
Theorem	itvid 28788	Utility theorem: index-independent form of df-itv 28784. (Contributed by Thierry Arnoux, 24-Aug-2017.)
⊢ Itv = Slot (Itv‘ndx)
Theorem	lngid 28789	Utility theorem: index-independent form of df-lng 28785. (Contributed by Thierry Arnoux, 27-Mar-2019.)
⊢ LineG = Slot (LineG‘ndx)
Theorem	slotsinbpsd 28790	The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 29340 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 28791	The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 29340 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 28792	The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 29339. (Contributed by AV, 9-Nov-2024.)
⊢ (LineG‘ndx) ≠ (Itv‘ndx)
Theorem	trkgstr 28793	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 28794	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 28795	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 28796	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 28797*	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 2782, 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 28798*	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 28799*	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 28800*	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)))}