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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnoseqno 28301 An element of a surreal sequence is a surreal. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑𝐴 No )    &   (𝜑𝐵𝑍)       (𝜑𝐵 No )
 
Theoremom2noseq0 28302 The mapping 𝐺 is a one-to-one mapping from ω onto a countable sequence of surreals that will be used to show the properties of seqs. This theorem shows the value of 𝐺 at ordinal zero. Compare the series of theorems starting at om2uz0i 13988. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))       (𝜑 → (𝐺‘∅) = 𝐶)
 
Theoremom2noseqsuc 28303* The value of 𝐺 at a successor. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺‘suc 𝐴) = ((𝐺𝐴) +s 1s ))
 
Theoremom2noseqfo 28304 Function statement for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       (𝜑𝐺:ω–onto𝑍)
 
Theoremom2noseqlt 28305* Surreal less-than relation for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴𝐵 → (𝐺𝐴) <s (𝐺𝐵)))
 
Theoremom2noseqlt2 28306* The mapping 𝐺 preserves order. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴𝐵 ↔ (𝐺𝐴) <s (𝐺𝐵)))
 
Theoremom2noseqf1o 28307* 𝐺 is a bijection. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       (𝜑𝐺:ω–1-1-onto𝑍)
 
Theoremom2noseqiso 28308* 𝐺 is an isomorphism from the finite ordinals to a surreal sequence. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       (𝜑𝐺 Isom E , <s (ω, 𝑍))
 
Theoremom2noseqoi 28309* An alternative definition of 𝐺 in terms of df-oi 9550. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))       (𝜑𝐺 = OrdIso( <s , 𝑍))
 
Theoremom2noseqrdg 28310* A helper lemma for the value of a recursive definition generator on a surreal sequence with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))       ((𝜑𝐵 ∈ ω) → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
 
Theoremnoseqrdglem 28311* A helper lemma for the value of a recursive defintion generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))       ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
 
Theoremnoseqrdgfn 28312* The recursive definition generator on surreal sequences is a function. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑𝑆 = ran 𝑅)       (𝜑𝑆 Fn 𝑍)
 
Theoremnoseqrdg0 28313* Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑𝑆 = ran 𝑅)       (𝜑 → (𝑆𝐶) = 𝐴)
 
Theoremnoseqrdgsuc 28314* Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑𝐶 No )    &   (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑𝑆 = ran 𝑅)       ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆𝐵)))
 
Theoremseqsfn 28315 The surreal sequence builder is a function. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑𝑀 No )    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))       (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)
 
Theoremseqs1 28316 The value of the surreal sequence bulder function at its initial value. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑𝑀 No )       (𝜑 → (seqs𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
 
Theoremseqsp1 28317 The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑𝑀 No )    &   (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))    &   (𝜑𝑁𝑍)       (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
 
15.6.3  Natural numbers
 
Syntaxcnn0s 28318 Declare the syntax for surreal non-negative integers.
class 0s
 
Syntaxcnns 28319 Declare the syntax for surreal positive integers.
class s
 
Definitiondf-n0s 28320 Define the set of non-negative surreal integers. This set behaves similarly to ω and 0, but it is a set of surreal numbers. Like those two sets, it satisfies the Peano axioms and is closed under (surreal) addition and multiplication. Compare df-nn 12267. (Contributed by Scott Fenton, 17-Mar-2025.)
0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
 
Definitiondf-nns 28321 Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
s = (ℕ0s ∖ { 0s })
 
Theoremn0sex 28322 The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
0s ∈ V
 
Theoremnnsex 28323 The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
s ∈ V
 
Theorempeano5n0s 28324* Peano's inductive postulate for non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
(( 0s𝐴 ∧ ∀𝑥𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s𝐴)
 
Theoremn0ssno 28325 The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
0s No
 
Theoremnnssn0s 28326 The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
s ⊆ ℕ0s
 
Theoremnnssno 28327 The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
s No
 
Theoremn0sno 28328 A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕ0s𝐴 No )
 
Theoremnnsno 28329 A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs𝐴 No )
 
Theoremn0snod 28330 A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝜑𝐴 ∈ ℕ0s)       (𝜑𝐴 No )
 
Theoremnnsnod 28331 A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝜑𝐴 ∈ ℕs)       (𝜑𝐴 No )
 
Theoremnnn0s 28332 A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕs𝐴 ∈ ℕ0s)
 
Theoremnnn0sd 28333 A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑𝐴 ∈ ℕs)       (𝜑𝐴 ∈ ℕ0s)
 
Theorem0n0s 28334 Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
0s ∈ ℕ0s
 
Theorempeano2n0s 28335 Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)
 
Theoremdfn0s2 28336* Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
0s = {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
 
Theoremn0sind 28337* Principle of Mathematical Induction (inference schema). Compare nnind 12284 and finds 7918. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝑥 = 0s → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 +s 1s ) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ0s → (𝜒𝜃))       (𝐴 ∈ ℕ0s𝜏)
 
Theoremn0scut 28338 A cut form for surreal naturals. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ ℕ0s𝐴 = ({(𝐴 -s 1s )} |s ∅))
 
Theoremn0ons 28339 A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ ℕ0s𝐴 ∈ Ons)
 
Theoremnnne0s 28340 A surreal positive integer is non-zero. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs𝐴 ≠ 0s )
 
Theoremn0sge0 28341 A non-negative integer is greater than or equal to zero. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
 
Theoremnnsgt0 28342 A positive integer is greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs → 0s <s 𝐴)
 
Theoremelnns 28343 Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))
 
Theoremelnns2 28344 A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
 
Theoremn0s0suc 28345* A non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-Jul-2025.)
(𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
 
Theoremnnsge1 28346 A positive surreal integer is greater than or equal to one. (Contributed by Scott Fenton, 26-Jul-2025.)
(𝑁 ∈ ℕs → 1s ≤s 𝑁)
 
Theoremn0addscl 28347 The non-negative surreal integers are closed under addition. (Contributed by Scott Fenton, 15-Apr-2025.)
((𝐴 ∈ ℕ0s𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)
 
Theoremn0mulscl 28348 The non-negative surreal integers are closed under multiplication. (Contributed by Scott Fenton, 15-Apr-2025.)
((𝐴 ∈ ℕ0s𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)
 
Theoremnnaddscl 28349 The positive surreal integers are closed under addition. (Contributed by Scott Fenton, 15-Apr-2025.)
((𝐴 ∈ ℕs𝐵 ∈ ℕs) → (𝐴 +s 𝐵) ∈ ℕs)
 
Theoremnnmulscl 28350 The positive surreal integers are closed under multiplication. (Contributed by Scott Fenton, 15-Apr-2025.)
((𝐴 ∈ ℕs𝐵 ∈ ℕs) → (𝐴 ·s 𝐵) ∈ ℕs)
 
Theorem1n0s 28351 Surreal one is a non-negative surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
1s ∈ ℕ0s
 
Theorem1nns 28352 Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
1s ∈ ℕs
 
Theorempeano2nns 28353 Peano postulate for positive surreal integers. One plus a positive surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
(𝐴 ∈ ℕs → (𝐴 +s 1s ) ∈ ℕs)
 
Theoremn0sbday 28354 A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝐴 ∈ ℕ0s → ( bday 𝐴) ∈ ω)
 
Theoremn0ssold 28355 The non-negative surreal integers are a subset of the old set of ω. (Contributed by Scott Fenton, 18-Apr-2025.)
0s ⊆ ( O ‘ω)
 
Theoremnnsrecgt0d 28356 The reciprocal of a positive surreal integer is positive. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑𝐴 ∈ ℕs)       (𝜑 → 0s <s ( 1s /su 𝐴))
 
Theoremseqn0sfn 28357 The surreal sequence builder is a function over 0s when started from zero. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → seqs 0s ( + , 𝐹) Fn ℕ0s)
 
Theoremeln0s 28358 A non-negative surreal integer is zero or a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs𝐴 = 0s ))
 
Theoremn0s0m1 28359 Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))
 
Theoremn0subs 28360 Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 26-May-2025.)
((𝑀 ∈ ℕ0s𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
 
Theoremn0p1nns 28361 One plus a non-negative surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
 
Theoremdfnns2 28362 Alternate definition of the positive surreal integers. Compare df-nn 12267. (Contributed by Scott Fenton, 6-Aug-2025.)
s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
 
Theoremnnsind 28363* Principle of Mathematical Induction (inference schema). (Contributed by Scott Fenton, 6-Aug-2025.)
(𝑥 = 1s → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 +s 1s ) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕs → (𝜒𝜃))       (𝐴 ∈ ℕs𝜏)
 
15.6.4  Integers
 
Syntaxczs 28364 Declare the syntax for surreal integers.
class s
 
Definitiondf-zs 28365 Define the surreal integers. Compare dfz2 12632. (Contributed by Scott Fenton, 17-May-2025.)
s = ( -s “ (ℕs × ℕs))
 
Theoremzsex 28366 The surreal integers form a set. (Contributed by Scott Fenton, 17-May-2025.)
s ∈ V
 
Theoremzssno 28367 The surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-May-2025.)
s No
 
Theoremzno 28368 A surreal integer is a surreal. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℤs𝐴 No )
 
Theoremznod 28369 A surreal integer is a surreal. Deduction form. (Contributed by Scott Fenton, 17-May-2025.)
(𝜑𝐴 ∈ ℤs)       (𝜑𝐴 No )
 
Theoremelzs 28370* Membership in the set of surreal integers. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
 
Theoremnnzsubs 28371 The difference of two surreal positive integers is an integer. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝐴 ∈ ℕs𝐵 ∈ ℕs) → (𝐴 -s 𝐵) ∈ ℤs)
 
Theoremnnzs 28372 A positive surreal integer is a surreal integer. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℕs𝐴 ∈ ℤs)
 
Theoremnnzsd 28373 A positive surreal integer is a surreal integer. Deduction form. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑𝐴 ∈ ℕs)       (𝜑𝐴 ∈ ℤs)
 
Theorem0zs 28374 Zero is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
0s ∈ ℤs
 
Theoremn0zs 28375 A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s𝐴 ∈ ℤs)
 
Theoremn0zsd 28376 A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑𝐴 ∈ ℕ0s)       (𝜑𝐴 ∈ ℤs)
 
Theorem1zs 28377 One is a surreal integer. (Contributed by Scott Fenton, 24-Jul-2025.)
1s ∈ ℤs
 
Theoremznegscl 28378 The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs → ( -us𝐴) ∈ ℤs)
 
Theoremznegscld 28379 The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑𝐴 ∈ ℤs)       (𝜑 → ( -us𝐴) ∈ ℤs)
 
Theoremzaddscl 28380 The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝐴 ∈ ℤs𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)
 
Theoremzaddscld 28381 The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑𝐴 ∈ ℤs)    &   (𝜑𝐵 ∈ ℤs)       (𝜑 → (𝐴 +s 𝐵) ∈ ℤs)
 
Theoremzsubscld 28382 The surreal integers are closed under subtraction. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑𝐴 ∈ ℤs)    &   (𝜑𝐵 ∈ ℤs)       (𝜑 → (𝐴 -s 𝐵) ∈ ℤs)
 
Theoremzmulscld 28383 The surreal integers are closed under multiplication. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝜑𝐴 ∈ ℤs)    &   (𝜑𝐵 ∈ ℤs)       (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)
 
Theoremelzn0s 28384 A surreal integer is a surreal that is a non-negative integer or whose negative is a non-negative integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs ↔ (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)))
 
Theoremelzs2 28385 A surreal integer is either a positive integer, zero, or the negative of a positive integer. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)))
 
Theoremeln0zs 28386 Non-negative surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
 
Theoremelnnzs 28387 Positive surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))
 
Theoremelznns 28388 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 ∨ ( -us𝑁) ∈ ℕ0s)))
 
Theoremzn0subs 28389 The non-negative difference of surreal integers is a non-negative integer. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝑀 ∈ ℤs𝑁 ∈ ℤs) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
 
Theorempeano5uzs 28390* Peano's inductive postulate for upper surreal integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑𝑁 ∈ ℤs)    &   (𝜑𝑁𝐴)    &   ((𝜑𝑥𝐴) → (𝑥 +s 1s ) ∈ 𝐴)       (𝜑 → {𝑘 ∈ ℤs𝑁 ≤s 𝑘} ⊆ 𝐴)
 
Theoremuzsind 28391* Induction on the upper surreal integers that start at 𝑀. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 +s 1s ) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤs𝜓)    &   ((𝑀 ∈ ℤs𝑘 ∈ ℤs𝑀 ≤s 𝑘) → (𝜒𝜃))       ((𝑀 ∈ ℤs𝑁 ∈ ℤs𝑀 ≤s 𝑁) → 𝜏)
 
Theoremzsbday 28392 A surreal integer has a finite birthday. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs → ( bday 𝐴) ∈ ω)
 
Theoremzscut 28393 A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝐴 ∈ ℤs𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
 
15.6.5  Dyadic fractions
 
Syntaxc2s 28394 Declare the syntax for surreal two.
class 2s
 
Definitiondf-2s 28395 Define surreal two. This is the simplest number greater than one. See 1p1e2s 28400 for its addition version. (Contributed by Scott Fenton, 27-May-2025.)
2s = ({ 1s } |s ∅)
 
Syntaxcexps 28396 Declare the syntax for surreal exponentiation.
class s
 
Definitiondf-exps 28397* Define surreal exponentiation. Compare df-exp 14103. (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}))‘( -us𝑦))))))
 
Syntaxczs12 28398 Define the syntax for the set of surreal dyadic fractions.
class s[1/2]
 
Definitiondf-zs12 28399* 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.)
s[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2ss𝑧))}
 
Theorem1p1e2s 28400 One plus one is two. Surreal version. (Contributed by Scott Fenton, 27-May-2025.)
( 1s +s 1s ) = 2s
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