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Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremon2ind 34201* Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
(π‘Ž = 𝑐 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑑 β†’ (πœ“ ↔ πœ’))    &   (π‘Ž = 𝑐 β†’ (πœƒ ↔ πœ’))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜏))    &   (𝑏 = π‘Œ β†’ (𝜏 ↔ πœ‚))    &   ((π‘Ž ∈ On ∧ 𝑏 ∈ On) β†’ ((βˆ€π‘ ∈ π‘Ž βˆ€π‘‘ ∈ 𝑏 πœ’ ∧ βˆ€π‘ ∈ π‘Ž πœ“ ∧ βˆ€π‘‘ ∈ 𝑏 πœƒ) β†’ πœ‘))    β‡’   ((𝑋 ∈ On ∧ π‘Œ ∈ On) β†’ πœ‚)
 
Theoremon3ind 34202* Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
(π‘Ž = 𝑑 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑒 β†’ (πœ“ ↔ πœ’))    &   (𝑐 = 𝑓 β†’ (πœ’ ↔ πœƒ))    &   (π‘Ž = 𝑑 β†’ (𝜏 ↔ πœƒ))    &   (𝑏 = 𝑒 β†’ (πœ‚ ↔ 𝜏))    &   (𝑏 = 𝑒 β†’ (𝜁 ↔ πœƒ))    &   (𝑐 = 𝑓 β†’ (𝜎 ↔ 𝜏))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜌))    &   (𝑏 = π‘Œ β†’ (𝜌 ↔ πœ‡))    &   (𝑐 = 𝑍 β†’ (πœ‡ ↔ πœ†))    &   ((π‘Ž ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) β†’ (((βˆ€π‘‘ ∈ π‘Ž βˆ€π‘’ ∈ 𝑏 βˆ€π‘“ ∈ 𝑐 πœƒ ∧ βˆ€π‘‘ ∈ π‘Ž βˆ€π‘’ ∈ 𝑏 πœ’ ∧ βˆ€π‘‘ ∈ π‘Ž βˆ€π‘“ ∈ 𝑐 𝜁) ∧ (βˆ€π‘‘ ∈ π‘Ž πœ“ ∧ βˆ€π‘’ ∈ 𝑏 βˆ€π‘“ ∈ 𝑐 𝜏 ∧ βˆ€π‘’ ∈ 𝑏 𝜎) ∧ βˆ€π‘“ ∈ 𝑐 πœ‚) β†’ πœ‘))    β‡’   ((𝑋 ∈ On ∧ π‘Œ ∈ On ∧ 𝑍 ∈ On) β†’ πœ†)
 
Theoremcoflton 34203* Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐡 and 𝐡 precedes 𝐢, then 𝐴 precedes 𝐢. Compare cofsslt 34228 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(πœ‘ β†’ 𝐴 βŠ† On)    &   (πœ‘ β†’ 𝐡 βŠ† On)    &   (πœ‘ β†’ 𝐢 βŠ† On)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 π‘₯ βŠ† 𝑦)    &   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 βˆ€π‘€ ∈ 𝐢 𝑧 ∈ 𝑀)    β‡’   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐴 βˆ€π‘ ∈ 𝐢 π‘Ž ∈ 𝑐)
 
Theoremcofon1 34204* Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐡 and the upper bound of 𝐴 dominates 𝐡, then their upper bounds are equal. Compare with cofcut1 34230 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ 𝒫 On)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 π‘₯ βŠ† 𝑦)    &   (πœ‘ β†’ 𝐡 βŠ† ∩ {𝑧 ∈ On ∣ 𝐴 βŠ† 𝑧})    β‡’   (πœ‘ β†’ ∩ {𝑧 ∈ On ∣ 𝐴 βŠ† 𝑧} = ∩ {𝑀 ∈ On ∣ 𝐡 βŠ† 𝑀})
 
Theoremcofon2 34205* Cofinality theorem for ordinals. If 𝐴 and 𝐡 are mutually cofinal, then their upper bounds agree. Compare cofcut2 34231 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ 𝒫 On)    &   (πœ‘ β†’ 𝐡 ∈ 𝒫 On)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 π‘₯ βŠ† 𝑦)    &   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)    β‡’   (πœ‘ β†’ ∩ {π‘Ž ∈ On ∣ 𝐴 βŠ† π‘Ž} = ∩ {𝑏 ∈ On ∣ 𝐡 βŠ† 𝑏})
 
Theoremcofonr 34206* Inverse cofinality law for ordinals. Contrast with cofcutr 34232 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ 𝐴 = ∩ {π‘₯ ∈ On ∣ 𝑋 βŠ† π‘₯})    β‡’   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘§ ∈ 𝑋 𝑦 βŠ† 𝑧)
 
Theoremnaddfn 34207 Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no Fn (On Γ— On)
 
Theoremnaddcllem 34208* Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((𝐴 +no 𝐡) ∈ On ∧ (𝐴 +no 𝐡) = ∩ {π‘₯ ∈ On ∣ (( +no β€œ ({𝐴} Γ— 𝐡)) βŠ† π‘₯ ∧ ( +no β€œ (𝐴 Γ— {𝐡})) βŠ† π‘₯)}))
 
Theoremnaddcl 34209 Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +no 𝐡) ∈ On)
 
Theoremnaddov 34210* The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +no 𝐡) = ∩ {π‘₯ ∈ On ∣ (( +no β€œ ({𝐴} Γ— 𝐡)) βŠ† π‘₯ ∧ ( +no β€œ (𝐴 Γ— {𝐡})) βŠ† π‘₯)})
 
Theoremnaddov2 34211* Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +no 𝐡) = ∩ {π‘₯ ∈ On ∣ (βˆ€π‘¦ ∈ 𝐡 (𝐴 +no 𝑦) ∈ π‘₯ ∧ βˆ€π‘§ ∈ 𝐴 (𝑧 +no 𝐡) ∈ π‘₯)})
 
Theoremnaddov3 34212* Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +no 𝐡) = ∩ {π‘₯ ∈ On ∣ (( +no β€œ ({𝐴} Γ— 𝐡)) βˆͺ ( +no β€œ (𝐴 Γ— {𝐡}))) βŠ† π‘₯})
 
Theoremnaddf 34213 Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
+no :(On Γ— On)⟢On
 
Theoremnaddcom 34214 Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +no 𝐡) = (𝐡 +no 𝐴))
 
Theoremnaddid1 34215 Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝐴 ∈ On β†’ (𝐴 +no βˆ…) = 𝐴)
 
Theoremnaddssim 34216 Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 βŠ† 𝐡 β†’ (𝐴 +no 𝐢) βŠ† (𝐡 +no 𝐢)))
 
Theoremnaddelim 34217 Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 ∈ 𝐡 β†’ (𝐴 +no 𝐢) ∈ (𝐡 +no 𝐢)))
 
Theoremnaddel1 34218 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 ∈ 𝐡 ↔ (𝐴 +no 𝐢) ∈ (𝐡 +no 𝐢)))
 
Theoremnaddel2 34219 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 ∈ 𝐡 ↔ (𝐢 +no 𝐴) ∈ (𝐢 +no 𝐡)))
 
Theoremnaddss1 34220 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 βŠ† 𝐡 ↔ (𝐴 +no 𝐢) βŠ† (𝐡 +no 𝐢)))
 
Theoremnaddss2 34221 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 βŠ† 𝐡 ↔ (𝐢 +no 𝐴) βŠ† (𝐢 +no 𝐡)))
 
Theoremnaddword1 34222 Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 βŠ† (𝐴 +no 𝐡))
 
Theoremnaddunif 34223* Uniformity theorem for natural addition. If 𝐴 is the upper bound of 𝑋 and 𝐡 is the upper bound of π‘Œ, then (𝐴 +no 𝐡) can be expressed in terms of 𝑋 and π‘Œ. (Contributed by Scott Fenton, 20-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ 𝐡 ∈ On)    &   (πœ‘ β†’ 𝐴 = ∩ {π‘₯ ∈ On ∣ 𝑋 βŠ† π‘₯})    &   (πœ‘ β†’ 𝐡 = ∩ {𝑦 ∈ On ∣ π‘Œ βŠ† 𝑦})    β‡’   (πœ‘ β†’ (𝐴 +no 𝐡) = ∩ {𝑧 ∈ On ∣ (( +no β€œ (𝑋 Γ— {𝐡})) βˆͺ ( +no β€œ ({𝐴} Γ— π‘Œ))) βŠ† 𝑧})
 
Theoremnaddasslem1 34224* Lemma for naddass 34226. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ ((𝐴 +no 𝐡) +no 𝐢) = ∩ {π‘₯ ∈ On ∣ (βˆ€π‘Ž ∈ 𝐴 ((π‘Ž +no 𝐡) +no 𝐢) ∈ π‘₯ ∧ βˆ€π‘ ∈ 𝐡 ((𝐴 +no 𝑏) +no 𝐢) ∈ π‘₯ ∧ βˆ€π‘ ∈ 𝐢 ((𝐴 +no 𝐡) +no 𝑐) ∈ π‘₯)})
 
Theoremnaddasslem2 34225* Lemma for naddass 34226. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ (𝐴 +no (𝐡 +no 𝐢)) = ∩ {π‘₯ ∈ On ∣ (βˆ€π‘Ž ∈ 𝐴 (π‘Ž +no (𝐡 +no 𝐢)) ∈ π‘₯ ∧ βˆ€π‘ ∈ 𝐡 (𝐴 +no (𝑏 +no 𝐢)) ∈ π‘₯ ∧ βˆ€π‘ ∈ 𝐢 (𝐴 +no (𝐡 +no 𝑐)) ∈ π‘₯)})
 
Theoremnaddass 34226 Natural ordinal addition is associative. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ ((𝐴 +no 𝐡) +no 𝐢) = (𝐴 +no (𝐡 +no 𝐢)))
 
Theoremnadd32 34227 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝐢 ∈ On) β†’ ((𝐴 +no 𝐡) +no 𝐢) = ((𝐴 +no 𝐢) +no 𝐡))
 
21.9.19  Surreal numbers: Cofinality and coinitiality
 
Theoremcofsslt 34228* If every element of 𝐴 is bounded by some element of 𝐡 and 𝐡 precedes 𝐢, then 𝐴 precedes 𝐢. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐡. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐡. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐴 ∈ 𝒫 No ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 π‘₯ ≀s 𝑦 ∧ 𝐡 <<s 𝐢) β†’ 𝐴 <<s 𝐢)
 
Theoremcoinitsslt 34229* If 𝐡 is coinitial with 𝐢 and 𝐴 precedes 𝐢, then 𝐴 precedes 𝐡. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐡 ∈ 𝒫 No ∧ βˆ€π‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐢 𝑦 ≀s π‘₯ ∧ 𝐴 <<s 𝐢) β†’ 𝐴 <<s 𝐡)
 
Theoremcofcut1 34230* If 𝐢 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐡 and the cut of 𝐴 and 𝐡 lies between 𝐢 and 𝐷. Then the cut of 𝐢 and 𝐷 is equal to the cut of 𝐴 and 𝐡. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐡 ∧ (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐢 π‘₯ ≀s 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 βˆƒπ‘€ ∈ 𝐷 𝑀 ≀s 𝑧) ∧ (𝐢 <<s {(𝐴 |s 𝐡)} ∧ {(𝐴 |s 𝐡)} <<s 𝐷)) β†’ (𝐴 |s 𝐡) = (𝐢 |s 𝐷))
 
Theoremcofcut2 34231* If 𝐴 and 𝐢 are mutually cofinal and 𝐡 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐡 is equal to the cut of 𝐢 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
(((𝐴 <<s 𝐡 ∧ 𝐢 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐢 π‘₯ ≀s 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 βˆƒπ‘€ ∈ 𝐷 𝑀 ≀s 𝑧) ∧ (βˆ€π‘‘ ∈ 𝐢 βˆƒπ‘’ ∈ 𝐴 𝑑 ≀s 𝑒 ∧ βˆ€π‘Ÿ ∈ 𝐷 βˆƒπ‘  ∈ 𝐡 𝑠 ≀s π‘Ÿ)) β†’ (𝐴 |s 𝐡) = (𝐢 |s 𝐷))
 
Theoremcofcutr 34232* If 𝑋 is the cut of 𝐴 and 𝐡, then 𝐴 is cofinal with ( L β€˜π‘‹) and 𝐡 is coinitial with ( R β€˜π‘‹). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐡 ∧ 𝑋 = (𝐴 |s 𝐡)) β†’ (βˆ€π‘₯ ∈ ( L β€˜π‘‹)βˆƒπ‘¦ ∈ 𝐴 π‘₯ ≀s 𝑦 ∧ βˆ€π‘§ ∈ ( R β€˜π‘‹)βˆƒπ‘€ ∈ 𝐡 𝑀 ≀s 𝑧))
 
Theoremcofcutrtime 34233* If 𝑋 is the cut of 𝐴 and 𝐡 and all of 𝐴 and 𝐡 are older than 𝑋, then ( L β€˜π‘‹) is cofinal with 𝐴 and ( R β€˜π‘‹) is coinitial with 𝐡. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
(((𝐴 βˆͺ 𝐡) βŠ† ( O β€˜( bday β€˜π‘‹)) ∧ 𝐴 <<s 𝐡 ∧ 𝑋 = (𝐴 |s 𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ( L β€˜π‘‹)π‘₯ ≀s 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 βˆƒπ‘€ ∈ ( R β€˜π‘‹)𝑀 ≀s 𝑧))
 
21.9.20  Surreal numbers: Induction and recursion on one variable
 
Syntaxcnorec 34234 Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
 
Definitiondf-norec 34235* Define the recursion generator for surreal functions of one variable. This generator creates a recursive function of surreals from their value on their left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
norec (𝐹) = frecs({⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}, No , 𝐹)
 
Theoremlrrecval 34236* The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴𝑅𝐡 ↔ 𝐴 ∈ (( L β€˜π΅) βˆͺ ( R β€˜π΅))))
 
Theoremlrrecval2 34237* Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴𝑅𝐡 ↔ ( bday β€˜π΄) ∈ ( bday β€˜π΅)))
 
Theoremlrrecpo 34238* Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   π‘… Po No
 
Theoremlrrecse 34239* Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   π‘… Se No
 
Theoremlrrecfr 34240* Now we show that 𝑅 is founded over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   π‘… Fr No
 
Theoremlrrecpred 34241* Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ π‘₯ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))}    β‡’   (𝐴 ∈ No β†’ Pred(𝑅, No , 𝐴) = (( L β€˜π΄) βˆͺ ( R β€˜π΄)))
 
Theoremnoinds 34242* Induction principle for a single surreal. If a property passes from a surreal's left and right sets to the surreal itself, then it holds for all surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ’))    &   (π‘₯ ∈ No β†’ (βˆ€π‘¦ ∈ (( L β€˜π‘₯) βˆͺ ( R β€˜π‘₯))πœ“ β†’ πœ‘))    β‡’   (𝐴 ∈ No β†’ πœ’)
 
Theoremnorecfn 34243 Surreal recursion over one variable is a function over the surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    β‡’   πΉ Fn No
 
Theoremnorecov 34244 Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    β‡’   (𝐴 ∈ No β†’ (πΉβ€˜π΄) = (𝐴𝐺(𝐹 β†Ύ (( L β€˜π΄) βˆͺ ( R β€˜π΄)))))
 
21.9.21  Surreal numbers: Induction and recursion on two variables
 
Syntaxcnorec2 34245 Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
 
Definitiondf-norec2 34246* Define surreal recursion on two variables. This function is key to the development of most of surreal numbers. (Contributed by Scott Fenton, 20-Aug-2024.)
norec2 (𝐹) = frecs({βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž ∈ ( No Γ— No ) ∧ 𝑏 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘Ž){βŸ¨π‘, π‘‘βŸ© ∣ 𝑐 ∈ (( L β€˜π‘‘) βˆͺ ( R β€˜π‘‘))} (1st β€˜π‘) ∨ (1st β€˜π‘Ž) = (1st β€˜π‘)) ∧ ((2nd β€˜π‘Ž){βŸ¨π‘, π‘‘βŸ© ∣ 𝑐 ∈ (( L β€˜π‘‘) βˆͺ ( R β€˜π‘‘))} (2nd β€˜π‘) ∨ (2nd β€˜π‘Ž) = (2nd β€˜π‘)) ∧ π‘Ž β‰  𝑏))}, ( No Γ— No ), 𝐹)
 
Theoremnoxpordpo 34247* To get through most of the textbook defintions in surreal numbers we will need recursion on two variables. This set of theorems sets up the preconditions for double recursion. This theorem establishes the partial ordering. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {βŸ¨π‘Ž, π‘βŸ© ∣ π‘Ž ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))}    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ( No Γ— No ) ∧ 𝑦 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑅(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   π‘† Po ( No Γ— No )
 
Theoremnoxpordfr 34248* Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {βŸ¨π‘Ž, π‘βŸ© ∣ π‘Ž ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))}    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ( No Γ— No ) ∧ 𝑦 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑅(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   π‘† Fr ( No Γ— No )
 
Theoremnoxpordse 34249* Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {βŸ¨π‘Ž, π‘βŸ© ∣ π‘Ž ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))}    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ( No Γ— No ) ∧ 𝑦 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑅(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   π‘† Se ( No Γ— No )
 
Theoremnoxpordpred 34250* Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {βŸ¨π‘Ž, π‘βŸ© ∣ π‘Ž ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))}    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ( No Γ— No ) ∧ 𝑦 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑅(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ Pred(𝑆, ( No Γ— No ), ⟨𝐴, 𝐡⟩) = ((((( L β€˜π΄) βˆͺ ( R β€˜π΄)) βˆͺ {𝐴}) Γ— ((( L β€˜π΅) βˆͺ ( R β€˜π΅)) βˆͺ {𝐡})) βˆ– {⟨𝐴, 𝐡⟩}))
 
Theoremno2indslem 34251* Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑅 = {βŸ¨π‘Ž, π‘βŸ© ∣ π‘Ž ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))}    &   π‘† = {βŸ¨π‘, π‘‘βŸ© ∣ (𝑐 ∈ ( No Γ— No ) ∧ 𝑑 ∈ ( No Γ— No ) ∧ (((1st β€˜π‘)𝑅(1st β€˜π‘‘) ∨ (1st β€˜π‘) = (1st β€˜π‘‘)) ∧ ((2nd β€˜π‘)𝑅(2nd β€˜π‘‘) ∨ (2nd β€˜π‘) = (2nd β€˜π‘‘)) ∧ 𝑐 β‰  𝑑))}    &   (π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝑀 β†’ (πœ“ ↔ πœ’))    &   (π‘₯ = 𝑧 β†’ (πœƒ ↔ πœ’))    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))    &   (𝑦 = 𝐡 β†’ (𝜏 ↔ πœ‚))    &   ((π‘₯ ∈ No ∧ 𝑦 ∈ No ) β†’ ((βˆ€π‘§ ∈ (( L β€˜π‘₯) βˆͺ ( R β€˜π‘₯))βˆ€π‘€ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))πœ’ ∧ βˆ€π‘§ ∈ (( L β€˜π‘₯) βˆͺ ( R β€˜π‘₯))πœ“ ∧ βˆ€π‘€ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))πœƒ) β†’ πœ‘))    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ πœ‚)
 
Theoremno2inds 34252* Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.)
(π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝑀 β†’ (πœ“ ↔ πœ’))    &   (π‘₯ = 𝑧 β†’ (πœƒ ↔ πœ’))    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))    &   (𝑦 = 𝐡 β†’ (𝜏 ↔ πœ‚))    &   ((π‘₯ ∈ No ∧ 𝑦 ∈ No ) β†’ ((βˆ€π‘§ ∈ (( L β€˜π‘₯) βˆͺ ( R β€˜π‘₯))βˆ€π‘€ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))πœ’ ∧ βˆ€π‘§ ∈ (( L β€˜π‘₯) βˆͺ ( R β€˜π‘₯))πœ“ ∧ βˆ€π‘€ ∈ (( L β€˜π‘¦) βˆͺ ( R β€˜π‘¦))πœƒ) β†’ πœ‘))    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ πœ‚)
 
Theoremnorec2fn 34253 The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    β‡’   πΉ Fn ( No Γ— No )
 
Theoremnorec2ov 34254 The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    β‡’   ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴𝐹𝐡) = (⟨𝐴, 𝐡⟩𝐺(𝐹 β†Ύ ((((( L β€˜π΄) βˆͺ ( R β€˜π΄)) βˆͺ {𝐴}) Γ— ((( L β€˜π΅) βˆͺ ( R β€˜π΅)) βˆͺ {𝐡})) βˆ– {⟨𝐴, 𝐡⟩}))))
 
Theoremno3inds 34255* Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.)
(π‘Ž = 𝑑 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑒 β†’ (πœ“ ↔ πœ’))    &   (𝑐 = 𝑓 β†’ (πœ’ ↔ πœƒ))    &   (π‘Ž = 𝑑 β†’ (𝜏 ↔ πœƒ))    &   (𝑏 = 𝑒 β†’ (πœ‚ ↔ 𝜏))    &   (𝑏 = 𝑒 β†’ (𝜁 ↔ πœƒ))    &   (𝑐 = 𝑓 β†’ (𝜎 ↔ 𝜏))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜌))    &   (𝑏 = π‘Œ β†’ (𝜌 ↔ πœ‡))    &   (𝑐 = 𝑍 β†’ (πœ‡ ↔ πœ†))    &   ((π‘Ž ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) β†’ (((βˆ€π‘‘ ∈ (( L β€˜π‘Ž) βˆͺ ( R β€˜π‘Ž))βˆ€π‘’ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))βˆ€π‘“ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))πœƒ ∧ βˆ€π‘‘ ∈ (( L β€˜π‘Ž) βˆͺ ( R β€˜π‘Ž))βˆ€π‘’ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))πœ’ ∧ βˆ€π‘‘ ∈ (( L β€˜π‘Ž) βˆͺ ( R β€˜π‘Ž))βˆ€π‘“ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))𝜁) ∧ (βˆ€π‘‘ ∈ (( L β€˜π‘Ž) βˆͺ ( R β€˜π‘Ž))πœ“ ∧ βˆ€π‘’ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))βˆ€π‘“ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))𝜏 ∧ βˆ€π‘’ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))𝜎) ∧ βˆ€π‘“ ∈ (( L β€˜π‘) βˆͺ ( R β€˜π‘))πœ‚) β†’ πœ‘))    β‡’   ((𝑋 ∈ No ∧ π‘Œ ∈ No ∧ 𝑍 ∈ No ) β†’ πœ†)
 
21.9.22  Surreal numbers - addition
 
Syntaxcadds 34256 Declare the syntax for surreal addition.
class +s
 
Definitiondf-adds 34257* Define surreal addition. This is the first of the field operations on the surreals. Definition from [Conway] p. 5. Definition from [Gonshor] p. 13. (Contributed by Scott Fenton, 20-Aug-2024.)
+s = norec2 ((π‘₯ ∈ V, π‘Ž ∈ V ↦ (({𝑦 ∣ βˆƒπ‘™ ∈ ( L β€˜(1st β€˜π‘₯))𝑦 = (π‘™π‘Ž(2nd β€˜π‘₯))} βˆͺ {𝑧 ∣ βˆƒπ‘™ ∈ ( L β€˜(2nd β€˜π‘₯))𝑧 = ((1st β€˜π‘₯)π‘Žπ‘™)}) |s ({𝑦 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜(1st β€˜π‘₯))𝑦 = (π‘Ÿπ‘Ž(2nd β€˜π‘₯))} βˆͺ {𝑧 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜(2nd β€˜π‘₯))𝑧 = ((1st β€˜π‘₯)π‘Žπ‘Ÿ)}))))
 
Theoremaddsfn 34258 Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
+s Fn ( No Γ— No )
 
Theoremaddsval 34259* The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴 +s 𝐡) = (({𝑦 ∣ βˆƒπ‘™ ∈ ( L β€˜π΄)𝑦 = (𝑙 +s 𝐡)} βˆͺ {𝑧 ∣ βˆƒπ‘™ ∈ ( L β€˜π΅)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π΄)𝑦 = (π‘Ÿ +s 𝐡)} βˆͺ {𝑧 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π΅)𝑧 = (𝐴 +s π‘Ÿ)})))
 
Theoremaddsval2 34260* The value of surreal addition with different choices for each bound variable. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴 +s 𝐡) = (({𝑦 ∣ βˆƒπ‘™ ∈ ( L β€˜π΄)𝑦 = (𝑙 +s 𝐡)} βˆͺ {𝑧 ∣ βˆƒπ‘š ∈ ( L β€˜π΅)𝑧 = (𝐴 +s π‘š)}) |s ({𝑀 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π΄)𝑀 = (π‘Ÿ +s 𝐡)} βˆͺ {𝑑 ∣ βˆƒπ‘  ∈ ( R β€˜π΅)𝑑 = (𝐴 +s 𝑠)})))
 
Theoremaddsid1 34261 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No β†’ (𝐴 +s 0s ) = 𝐴)
 
Theoremaddsid1d 34262 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ No )    β‡’   (πœ‘ β†’ (𝐴 +s 0s ) = 𝐴)
 
Theoremaddscom 34263 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴 +s 𝐡) = (𝐡 +s 𝐴))
 
Theoremaddscomd 34264 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    β‡’   (πœ‘ β†’ (𝐴 +s 𝐡) = (𝐡 +s 𝐴))
 
Theoremaddsproplem1 34265* Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8313 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    &   (πœ‘ β†’ ((( bday β€˜π΄) +no ( bday β€˜π΅)) βˆͺ (( bday β€˜π΄) +no ( bday β€˜πΆ))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))))    β‡’   (πœ‘ β†’ ((𝐴 +s 𝐡) ∈ No ∧ (𝐡 <s 𝐢 β†’ (𝐡 +s 𝐴) <s (𝐢 +s 𝐴))))
 
Theoremaddsproplem2 34266* Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    β‡’   (πœ‘ β†’ ({𝑝 ∣ βˆƒπ‘™ ∈ ( L β€˜π‘‹)𝑝 = (𝑙 +s π‘Œ)} βˆͺ {π‘ž ∣ βˆƒπ‘š ∈ ( L β€˜π‘Œ)π‘ž = (𝑋 +s π‘š)}) <<s ({𝑀 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π‘‹)𝑀 = (π‘Ÿ +s π‘Œ)} βˆͺ {𝑑 ∣ βˆƒπ‘  ∈ ( R β€˜π‘Œ)𝑑 = (𝑋 +s 𝑠)}))
 
Theoremaddsproplem3 34267* Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    β‡’   (πœ‘ β†’ ((𝑋 +s π‘Œ) ∈ No ∧ ({𝑝 ∣ βˆƒπ‘™ ∈ ( L β€˜π‘‹)𝑝 = (𝑙 +s π‘Œ)} βˆͺ {π‘ž ∣ βˆƒπ‘š ∈ ( L β€˜π‘Œ)π‘ž = (𝑋 +s π‘š)}) <<s {(𝑋 +s π‘Œ)} ∧ {(𝑋 +s π‘Œ)} <<s ({𝑀 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π‘‹)𝑀 = (π‘Ÿ +s π‘Œ)} βˆͺ {𝑑 ∣ βˆƒπ‘  ∈ ( R β€˜π‘Œ)𝑑 = (𝑋 +s 𝑠)})))
 
Theoremaddsproplem4 34268* Lemma for surreal addition properties. Show the second half of the inductive hypothesis when π‘Œ is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    &   (πœ‘ β†’ 𝑍 ∈ No )    &   (πœ‘ β†’ π‘Œ <s 𝑍)    &   (πœ‘ β†’ ( bday β€˜π‘Œ) ∈ ( bday β€˜π‘))    β‡’   (πœ‘ β†’ (π‘Œ +s 𝑋) <s (𝑍 +s 𝑋))
 
Theoremaddsproplem5 34269* Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than π‘Œ. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    &   (πœ‘ β†’ 𝑍 ∈ No )    &   (πœ‘ β†’ π‘Œ <s 𝑍)    &   (πœ‘ β†’ ( bday β€˜π‘) ∈ ( bday β€˜π‘Œ))    β‡’   (πœ‘ β†’ (π‘Œ +s 𝑋) <s (𝑍 +s 𝑋))
 
Theoremaddsproplem6 34270* Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when π‘Œ and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    &   (πœ‘ β†’ 𝑍 ∈ No )    &   (πœ‘ β†’ π‘Œ <s 𝑍)    &   (πœ‘ β†’ ( bday β€˜π‘Œ) = ( bday β€˜π‘))    β‡’   (πœ‘ β†’ (π‘Œ +s 𝑋) <s (𝑍 +s 𝑋))
 
Theoremaddsproplem7 34271* Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ βˆ€π‘₯ ∈ No βˆ€π‘¦ ∈ No βˆ€π‘§ ∈ No (((( bday β€˜π‘₯) +no ( bday β€˜π‘¦)) βˆͺ (( bday β€˜π‘₯) +no ( bday β€˜π‘§))) ∈ ((( bday β€˜π‘‹) +no ( bday β€˜π‘Œ)) βˆͺ (( bday β€˜π‘‹) +no ( bday β€˜π‘))) β†’ ((π‘₯ +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 β†’ (𝑦 +s π‘₯) <s (𝑧 +s π‘₯)))))    &   (πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    &   (πœ‘ β†’ 𝑍 ∈ No )    &   (πœ‘ β†’ π‘Œ <s 𝑍)    β‡’   (πœ‘ β†’ (π‘Œ +s 𝑋) <s (𝑍 +s 𝑋))
 
Theoremaddsprop 34272 Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝑋 ∈ No ∧ π‘Œ ∈ No ∧ 𝑍 ∈ No ) β†’ ((𝑋 +s π‘Œ) ∈ No ∧ (π‘Œ <s 𝑍 β†’ (π‘Œ +s 𝑋) <s (𝑍 +s 𝑋))))
 
Theoremaddscut 34273* Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    β‡’   (πœ‘ β†’ ((𝑋 +s π‘Œ) ∈ No ∧ ({𝑝 ∣ βˆƒπ‘™ ∈ ( L β€˜π‘‹)𝑝 = (𝑙 +s π‘Œ)} βˆͺ {π‘ž ∣ βˆƒπ‘š ∈ ( L β€˜π‘Œ)π‘ž = (𝑋 +s π‘š)}) <<s {(𝑋 +s π‘Œ)} ∧ {(𝑋 +s π‘Œ)} <<s ({𝑀 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜π‘‹)𝑀 = (π‘Ÿ +s π‘Œ)} βˆͺ {𝑑 ∣ βˆƒπ‘  ∈ ( R β€˜π‘Œ)𝑑 = (𝑋 +s 𝑠)})))
 
Theoremaddscld 34274 Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ 𝑋 ∈ No )    &   (πœ‘ β†’ π‘Œ ∈ No )    β‡’   (πœ‘ β†’ (𝑋 +s π‘Œ) ∈ No )
 
Theoremaddscl 34275 Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (𝐴 +s 𝐡) ∈ No )
 
Theoremaddsf 34276 Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
+s :( No Γ— No )⟢ No
 
Theoremaddsfo 34277 Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
+s :( No Γ— No )–ontoβ†’ No
 
Theoremsltadd1im 34278 Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 <s 𝐡 β†’ (𝐴 +s 𝐢) <s (𝐡 +s 𝐢)))
 
Theoremsltadd2im 34279 Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 <s 𝐡 β†’ (𝐢 +s 𝐴) <s (𝐢 +s 𝐡)))
 
Theoremsleadd1im 34280 Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ ((𝐴 +s 𝐢) ≀s (𝐡 +s 𝐢) β†’ 𝐴 ≀s 𝐡))
 
Theoremsleadd2im 34281 Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ ((𝐢 +s 𝐴) ≀s (𝐢 +s 𝐡) β†’ 𝐴 ≀s 𝐡))
 
Theoremsleadd1 34282 Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 ≀s 𝐡 ↔ (𝐴 +s 𝐢) ≀s (𝐡 +s 𝐢)))
 
Theoremsleadd2 34283 Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 ≀s 𝐡 ↔ (𝐢 +s 𝐴) ≀s (𝐢 +s 𝐡)))
 
Theoremsltadd2 34284 Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 <s 𝐡 ↔ (𝐢 +s 𝐴) <s (𝐢 +s 𝐡)))
 
Theoremsltadd1 34285 Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ (𝐴 <s 𝐡 ↔ (𝐴 +s 𝐢) <s (𝐡 +s 𝐢)))
 
Theoremaddscan2 34286 Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ ((𝐴 +s 𝐢) = (𝐡 +s 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremaddscan1 34287 Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ ((𝐢 +s 𝐴) = (𝐢 +s 𝐡) ↔ 𝐴 = 𝐡))
 
Theoremaddsunif 34288* Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define 𝐴 and 𝐡 in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ 𝐿 <<s 𝑅)    &   (πœ‘ β†’ 𝑀 <<s 𝑆)    &   (πœ‘ β†’ 𝐴 = (𝐿 |s 𝑅))    &   (πœ‘ β†’ 𝐡 = (𝑀 |s 𝑆))    β‡’   (πœ‘ β†’ (𝐴 +s 𝐡) = (({𝑦 ∣ βˆƒπ‘™ ∈ 𝐿 𝑦 = (𝑙 +s 𝐡)} βˆͺ {𝑧 ∣ βˆƒπ‘š ∈ 𝑀 𝑧 = (𝐴 +s π‘š)}) |s ({𝑀 ∣ βˆƒπ‘Ÿ ∈ 𝑅 𝑀 = (π‘Ÿ +s 𝐡)} βˆͺ {𝑑 ∣ βˆƒπ‘  ∈ 𝑆 𝑑 = (𝐴 +s 𝑠)})))
 
Theoremaddsasslem1 34289* Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    β‡’   (πœ‘ β†’ ((𝐴 +s 𝐡) +s 𝐢) = ((({𝑦 ∣ βˆƒπ‘™ ∈ ( L β€˜π΄)𝑦 = ((𝑙 +s 𝐡) +s 𝐢)} βˆͺ {𝑧 ∣ βˆƒπ‘š ∈ ( L β€˜π΅)𝑧 = ((𝐴 +s π‘š) +s 𝐢)}) βˆͺ {𝑀 ∣ βˆƒπ‘› ∈ ( L β€˜πΆ)𝑀 = ((𝐴 +s 𝐡) +s 𝑛)}) |s (({π‘Ž ∣ βˆƒπ‘ ∈ ( R β€˜π΄)π‘Ž = ((𝑝 +s 𝐡) +s 𝐢)} βˆͺ {𝑏 ∣ βˆƒπ‘ž ∈ ( R β€˜π΅)𝑏 = ((𝐴 +s π‘ž) +s 𝐢)}) βˆͺ {𝑐 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜πΆ)𝑐 = ((𝐴 +s 𝐡) +s π‘Ÿ)})))
 
Theoremaddsasslem2 34290* Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    β‡’   (πœ‘ β†’ (𝐴 +s (𝐡 +s 𝐢)) = ((({𝑦 ∣ βˆƒπ‘™ ∈ ( L β€˜π΄)𝑦 = (𝑙 +s (𝐡 +s 𝐢))} βˆͺ {𝑧 ∣ βˆƒπ‘š ∈ ( L β€˜π΅)𝑧 = (𝐴 +s (π‘š +s 𝐢))}) βˆͺ {𝑀 ∣ βˆƒπ‘› ∈ ( L β€˜πΆ)𝑀 = (𝐴 +s (𝐡 +s 𝑛))}) |s (({π‘Ž ∣ βˆƒπ‘ ∈ ( R β€˜π΄)π‘Ž = (𝑝 +s (𝐡 +s 𝐢))} βˆͺ {𝑏 ∣ βˆƒπ‘ž ∈ ( R β€˜π΅)𝑏 = (𝐴 +s (π‘ž +s 𝐢))}) βˆͺ {𝑐 ∣ βˆƒπ‘Ÿ ∈ ( R β€˜πΆ)𝑐 = (𝐴 +s (𝐡 +s π‘Ÿ))})))
 
Theoremaddsass 34291 Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ 𝐢 ∈ No ) β†’ ((𝐴 +s 𝐡) +s 𝐢) = (𝐴 +s (𝐡 +s 𝐢)))
 
Theoremaddsassd 34292 Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    β‡’   (πœ‘ β†’ ((𝐴 +s 𝐡) +s 𝐢) = (𝐴 +s (𝐡 +s 𝐢)))
 
Theoremadds32d 34293 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐢 ∈ No )    β‡’   (πœ‘ β†’ ((𝐴 +s 𝐡) +s 𝐢) = ((𝐴 +s 𝐢) +s 𝐡))
 
21.9.23  Surreal numbers - negation and subtraction
 
Syntaxcnegs 34294 Declare the syntax for surreal negation.
class -us
 
Syntaxcsubs 34295 Declare the syntax for surreal subtraction.
class -s
 
Definitiondf-negs 34296* Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
-us = norec ((π‘₯ ∈ V, 𝑛 ∈ V ↦ ((𝑛 β€œ ( R β€˜π‘₯)) |s (𝑛 β€œ ( L β€˜π‘₯)))))
 
Definitiondf-subs 34297* Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
-s = (π‘₯ ∈ No , 𝑦 ∈ No ↦ (π‘₯ +s ( -us β€˜π‘¦)))
 
Theoremnegsfn 34298 Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
-us Fn No
 
Theoremnegsval 34299 The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No β†’ ( -us β€˜π΄) = (( -us β€œ ( R β€˜π΄)) |s ( -us β€œ ( L β€˜π΄))))
 
Theoremnegs0s 34300 Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
( -us β€˜ 0s ) = 0s
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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