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
Syntax	cnew 27901	Declare the symbol for the new on function.
class N
Syntax	cleft 27902	Declare the symbol for the left option function.
class L
Syntax	cright 27903	Declare the symbol for the right option function.
class R
Definition	df-made 27904	Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
Definition	df-old 27905	Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
Definition	df-new 27906	Define the newer than function. This function carries an ordinal to all surreals made on that day. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
Definition	df-left 27907*	Define the left options of a surreal. This is the set of surreals that are simpler and less than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
Definition	df-right 27908*	Define the right options of a surreal. This is the set of surreals that are simpler and greater than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
Theorem	madeval 27909	The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
Theorem	madeval2 27910*	Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
Theorem	oldval 27911	The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
Theorem	newval 27912	The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
Theorem	madef 27913	The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ M :On⟶𝒫 No
Theorem	oldf 27914	The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ O :On⟶𝒫 No
Theorem	newf 27915	The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ N :On⟶𝒫 No
Theorem	old0 27916	No surreal is older than ∅. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( O ‘∅) = ∅
Theorem	madessno 27917	Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( M ‘𝐴) ⊆ No
Theorem	oldssno 27918	Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ No
Theorem	newssno 27919	New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) ⊆ No
Theorem	leftval 27920*	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 27921*	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 27922	The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L : No ⟶𝒫 No
Theorem	rightf 27923	The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R : No ⟶𝒫 No
Theorem	elmade 27924*	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 27925*	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 27926*	Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
Theorem	ssltleft 27927	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 27928	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 27929	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 27930	The only surreal made on day ∅ is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( M ‘∅) = { 0s }
Theorem	new0 27931	The only surreal new on day ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ( N ‘∅) = { 0s }
Theorem	old1 27932	The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( O ‘1o) = { 0s }
Theorem	madess 27933	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 27934	The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
Theorem	leftssold 27935	The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	rightssold 27936	The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	leftssno 27937	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 27938	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 27939	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 27940	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 27941	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 27942	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 27943	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 27944	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 27945	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 27946	No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋))
Theorem	leftirr 27947	No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( L ‘𝑋)
Theorem	rightirr 27948	No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( R ‘𝑋)
Theorem	left0s 27949	The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( L ‘ 0s ) = ∅
Theorem	right0s 27950	The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( R ‘ 0s ) = ∅
Theorem	left1s 27951	The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( L ‘ 1s ) = { 0s }
Theorem	right1s 27952	The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( R ‘ 1s ) = ∅
Theorem	lrold 27953	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 27954*	Lemma for madebday 27956. If the inductive hypothesis of madebday 27956 is satisfied, the converse of oldbdayim 27945 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 27955*	Lemma for madebday 27956. If the inductive hypothesis of madebday 27956 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27959 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 27956	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 27957	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 27958	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 27959	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 27960	The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ |s : <<s –onto→ No
Theorem	sltn0 27961	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 27962	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 27963	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 27964	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 27965	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 27966	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 27967	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 27968	The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)
Theorem	oldfi 27969	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 27970*	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 27971*	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 27972*	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 27973*	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 27974*	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 27975*	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 27976*	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 27977*	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 27978*	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 27979*	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 27980*	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 27981*	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 27982*	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 27983*	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 27984*	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 27985*	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 27986*	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 27987*	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 27988	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
Definition	df-norec 27989*	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 27990*	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 27991*	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 27992*	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 27993*	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 27994*	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 27995*	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 27996*	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 27997	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 27998	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 27999	Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
Definition	df-norec2 28000*	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)