P. 280 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	leftssno 27901	The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝐴) ⊆ No
Theorem	rightssno 27902	The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝐴) ⊆ No
Theorem	madecut 27903	Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴))
Theorem	madeun 27904	The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))
Theorem	madeoldsuc 27905	The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
Theorem	oldsuc 27906	The value of the old set at a successor. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝐴 ∈ On → ( O ‘suc 𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)))
Theorem	oldlim 27907	The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → ( O ‘𝐴) = ∪ ( O “ 𝐴))
Theorem	madebdayim 27908	If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
Theorem	oldbdayim 27909	If 𝑋 is in the old set for 𝐴, then the birthday of 𝑋 is less than 𝐴. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)
Theorem	oldirr 27910	No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋))
Theorem	leftirr 27911	No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( L ‘𝑋)
Theorem	rightirr 27912	No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( R ‘𝑋)
Theorem	left0s 27913	The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( L ‘ 0s ) = ∅
Theorem	right0s 27914	The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( R ‘ 0s ) = ∅
Theorem	left1s 27915	The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( L ‘ 1s ) = { 0s }
Theorem	right1s 27916	The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( R ‘ 1s ) = ∅
Theorem	lrold 27917	The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
Theorem	madebdaylemold 27918*	Lemma for madebday 27920. If the inductive hypothesis of madebday 27920 is satisfied, the converse of oldbdayim 27909 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
Theorem	madebdaylemlrcut 27919*	Lemma for madebday 27920. If the inductive hypothesis of madebday 27920 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27923 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	madebday 27920	A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
Theorem	oldbday 27921	A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
Theorem	newbday 27922	A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))
Theorem	lrcut 27923	A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	scutfo 27924	The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ |s : <<s –onto→ No
Theorem	sltn0 27925	If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))
Theorem	lruneq 27926	If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
Theorem	sltlpss 27927	If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
Theorem	slelss 27928	If two surreals 𝐴 and 𝐵 share a birthday, then 𝐴 ≤s 𝐵 if and only if the left set of 𝐴 is a non-strict subset of the left set of 𝐵. (Contributed by Scott Fenton, 21-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵)))
Theorem	0elold 27929	Zero is in the old set of any non-zero number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≠ 0s )    ⇒   ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))
Theorem	0elleft 27930	Zero is in the left set of any positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 0s ∈ ( L ‘𝐴))
Theorem	0elright 27931	Zero is in the right set of any negative number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 0s )    ⇒   ⊢ (𝜑 → 0s ∈ ( R ‘𝐴))
15.3.4  Cofinality and coinitiality
Theorem	cofsslt 27932*	If every element of 𝐴 is bounded above by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Theorem	coinitsslt 27933*	If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
Theorem	cofcut1 27934*	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut1d 27935*	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)    &   ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)})    &   ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut2 27936*	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut2d 27937*	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 No )    &   ⊢ (𝜑 → 𝐷 ∈ 𝒫 No )    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)    &   ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢)    &   ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcutr 27938*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧))
Theorem	cofcutr1d 27939*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋). First half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)
Theorem	cofcutr2d 27940*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐵 is coinitial with ( R ‘𝑋). Second half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧)
Theorem	cofcutrtime 27941*	If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. 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 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
Theorem	cofcutrtime1d 27942*	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( L ‘𝑋) is cofinal with 𝐴. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)))    &   ⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
Theorem	cofcutrtime2d 27943*	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( R ‘𝑋) is coinitial with 𝐵. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)))    &   ⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
Theorem	cofss 27944*	Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)
Theorem	coiniss 27945*	Coinitiality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
Theorem	cutlt 27946*	Eliminating all elements below a given element of a cut does not affect the cut. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐿)    ⇒   ⊢ (𝜑 → 𝐴 = (({𝑋} ∪ {𝑦 ∈ 𝐿 ∣ 𝑋 <s 𝑦}) |s 𝑅))
Theorem	cutpos 27947*	Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴)))
15.4  Induction and recursion
15.4.1  Induction and recursion on one variable
Syntax	cnorec 27948	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
Definition	df-norec 27949*	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({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
Theorem	lrrecval 27950*	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 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
Theorem	lrrecval2 27951*	Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
Theorem	lrrecpo 27952*	Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ 𝑅 Po No
Theorem	lrrecse 27953*	Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ 𝑅 Se No
Theorem	lrrecfr 27954*	Now we show that 𝑅 is founded over No . (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ 𝑅 Fr No
Theorem	lrrecpred 27955*	Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
Theorem	noinds 27956*	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.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ No → (∀𝑦 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ No → 𝜒)
Theorem	norecfn 27957	Surreal recursion over one variable is a function over the surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝐹 = norec (𝐺)    ⇒   ⊢ 𝐹 Fn No
Theorem	norecov 27958	Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝐹 = norec (𝐺)    ⇒   ⊢ (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
15.4.2  Induction and recursion on two variables
Syntax	cnorec2 27959	Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
Definition	df-norec2 27960*	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({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹)
Theorem	noxpordpo 27961*	To get through most of the textbook definitions 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 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ 𝑆 Po ( No × No )
Theorem	noxpordfr 27962*	Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ 𝑆 Fr ( No × No )
Theorem	noxpordse 27963*	Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ 𝑆 Se ( No × No )
Theorem	noxpordpred 27964*	Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
Theorem	no2indslem 27965*	Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
Theorem	no2inds 27966*	Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
Theorem	norec2fn 27967	The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ 𝐹 = norec2 (𝐺)    ⇒   ⊢ 𝐹 Fn ( No × No )
Theorem	norec2ov 27968	The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ 𝐹 = norec2 (𝐺)    ⇒   ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
Theorem	no3inds 27969*	Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆)
15.5  Surreal arithmetic
15.5.1  Addition
Syntax	cadds 27970	Declare the syntax for surreal addition.
class +s
Definition	df-adds 27971*	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 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))))
Theorem	addsfn 27972	Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ +s Fn ( No × No )
Theorem	addsval 27973*	The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑧 = (𝐴 +s 𝑟)})))
Theorem	addsval2 27974*	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 ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)})))
Theorem	addsrid 27975	Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)
Theorem	addsridd 27976	Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 +s 0s ) = 𝐴)
Theorem	addscom 27977	Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
Theorem	addscomd 27978	Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
Theorem	addslid 27979	Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴)
Theorem	addsproplem1 27980*	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 8421 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.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (( bday ‘𝐴) +no ( bday ‘𝐶))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Theorem	addsproplem2 27981*	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.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))
Theorem	addsproplem3 27982*	Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addsproplem4 27983*	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem5 27984*	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem6 27985*	Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem7 27986*	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.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsprop 27987	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 ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))
Theorem	addscutlem 27988*	Lemma for addscut 27989. Show the statement with some additional distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addscut 27989*	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.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addscut2 27990*	Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))
Theorem	addscld 27991	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
Theorem	addscl 27992	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
Theorem	addsf 27993	Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +s :( No × No )⟶ No
Theorem	addsfo 27994	Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +s :( No × No )–onto→ No
Theorem	peano2no 27995	A theorem for surreals that is analogous to the second Peano postulate peano2 7894. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (𝐴 ∈ No → (𝐴 +s 1s ) ∈ No )
Theorem	sltadd1im 27996	Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐴 +s 𝐶) <s (𝐵 +s 𝐶)))
Theorem	sltadd2im 27997	Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐶 +s 𝐴) <s (𝐶 +s 𝐵)))
Theorem	sleadd1im 27998	Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵))
Theorem	sleadd2im 27999	Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵) → 𝐴 ≤s 𝐵))
Theorem	sleadd1 28000	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 ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))