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	newssno 27901	New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) ⊆ No
Theorem	leftval 27902*	The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
Theorem	rightval 27903*	The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
Theorem	leftf 27904	The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L : No ⟶𝒫 No
Theorem	rightf 27905	The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R : No ⟶𝒫 No
Theorem	elmade 27906*	Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
Theorem	elmade2 27907*	Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
Theorem	elold 27908*	Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
Theorem	ssltleft 27909	A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
Theorem	ssltright 27910	A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴))
Theorem	lltropt 27911	The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.) (Revised by Scott Fenton, 21-Feb-2025.)
⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
Theorem	made0 27912	The only surreal made on day ∅ is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( M ‘∅) = { 0s }
Theorem	new0 27913	The only surreal new on day ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ( N ‘∅) = { 0s }
Theorem	old1 27914	The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( O ‘1o) = { 0s }
Theorem	madess 27915	If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
Theorem	oldssmade 27916	The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
Theorem	leftssold 27917	The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	rightssold 27918	The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	leftssno 27919	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 27920	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 27921	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 27922	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 27923	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 27924	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 27925	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 27926	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 27927	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 27928	No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋))
Theorem	leftirr 27929	No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( L ‘𝑋)
Theorem	rightirr 27930	No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( R ‘𝑋)
Theorem	left0s 27931	The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( L ‘ 0s ) = ∅
Theorem	right0s 27932	The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( R ‘ 0s ) = ∅
Theorem	left1s 27933	The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( L ‘ 1s ) = { 0s }
Theorem	right1s 27934	The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( R ‘ 1s ) = ∅
Theorem	lrold 27935	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 27936*	Lemma for madebday 27938. If the inductive hypothesis of madebday 27938 is satisfied, the converse of oldbdayim 27927 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 27937*	Lemma for madebday 27938. If the inductive hypothesis of madebday 27938 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27941 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 27938	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 27939	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 27940	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 27941	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 27942	The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ |s : <<s –onto→ No
Theorem	sltn0 27943	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 27944	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 27945	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 27946	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 27947	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 27948	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 27949	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))
Theorem	madefi 27950	The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)
Theorem	oldfi 27951	The old set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
15.3.4  Cofinality and coinitiality
Theorem	cofsslt 27952*	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 27953*	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 27954*	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 27955*	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 27956*	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 27957*	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 27958*	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 27959*	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 27960*	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 27961*	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 27962*	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 27963*	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 27964*	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 27965*	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 27966*	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 27967*	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)))
Theorem	cutmax 27968*	If 𝐴 has a maximum, then the maximum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵))
Theorem	cutmin 27969*	If 𝐵 has a minimum, then the minimum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝑋 ≤s 𝑦)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐴 |s {𝑋}))
15.4  Induction and recursion
15.4.1  Induction and recursion on one variable
Syntax	cnorec 27970	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
Definition	df-norec 27971*	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 27972*	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 27973*	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 27974*	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 27975*	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 27976*	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 27977*	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 27978*	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 27979	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 27980	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 27981	Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
Definition	df-norec2 27982*	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 27983*	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 27984*	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 27985*	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 27986*	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 27987*	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 27988*	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 27989	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 27990	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 27991*	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 27992	Declare the syntax for surreal addition.
class +s
Definition	df-adds 27993*	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 27994	Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ +s Fn ( No × No )
Theorem	addsval 27995*	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 27996*	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 27997	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 27998	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 27999	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 28000	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))