P. 287 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	muls0ord 28601	If a surreal product is zero, one of its factors must be zero. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))
Theorem	mulsne0bd 28602	The product of two nonzero surreals is nonzero. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ≠ 0s ↔ (𝐴 ≠ 0s ∧ 𝐵 ≠ 0s )))
15.5.4  Division
Syntax	cdivs 28603	Declare the syntax for surreal division.
class /su
Definition	df-divs 28604*	Define surreal division. This is not the definition used in the literature, but we use it here because it is technically easier to work with. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ /su = (𝑥 ∈ No, 𝑦 ∈ (No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))
Theorem	divsval 28605*	The value of surreal division. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
Theorem	norecdiv 28606*	If a surreal has a reciprocal, then it has any division. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)
Theorem	noreceuw 28607*	If a surreal has a reciprocal, then it has unique division. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)
Theorem	recsne0 28608*	If a surreal has a reciprocal, then it is nonzero. (Contributed by Scott Fenton, 5-Sep-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0s )
Theorem	divmulsw 28609*	Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28634, we can eliminate the existence hypothesis (see divmuls 28637). (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
Theorem	divmulswd 28610*	Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
Theorem	divsclw 28611*	Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s ) → (𝐴 /su 𝐵) ∈ No)
Theorem	divsclwd 28612*	Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → (𝐴 /su 𝐵) ∈ No)
Theorem	divscan2wd 28613*	A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → (𝐵 ·s (𝐴 /su 𝐵)) = 𝐴)
Theorem	divscan1wd 28614*	A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐵) ·s 𝐵) = 𝐴)
Theorem	ltdivmulswd 28615*	Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐶 ·s 𝐵)))
Theorem	ltdivmuls2wd 28616*	Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 ·s 𝐶)))
Theorem	ltmuldivswd 28617*	Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
Theorem	ltmuldivs2wd 28618*	Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
Theorem	divsasswd 28619*	An associative law for surreal division. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    &   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶)))
Theorem	divs1 28620	A surreal divided by one is itself. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝐴 ∈ No → (𝐴 /su 1s ) = 𝐴)
Theorem	divs1d 28621	A surreal divided by one is itself. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 /su 1s ) = 𝐴)
Theorem	precsexlemcbv 28622*	Lemma for surreal reciprocals. Change some bound variables. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    ⇒   ⊢ 𝐹 = rec((𝑞 ∈ V ↦ ⦋(1st ‘𝑞) / 𝑚⦌⦋(2nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ (R‘𝐴)∃𝑤 ∈ 𝑚 𝑏 = (( 1s +s ((𝑧𝑅 −s 𝐴) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑧 ∈ (L‘𝐴) ∣ 0s <s 𝑧}∃𝑡 ∈ 𝑠 𝑏 = (( 1s +s ((𝑧𝐿 −s 𝐴) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑧 ∈ (L‘𝐴) ∣ 0s <s 𝑧}∃𝑤 ∈ 𝑚 𝑏 = (( 1s +s ((𝑧𝐿 −s 𝐴) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ (R‘𝐴)∃𝑡 ∈ 𝑠 𝑏 = (( 1s +s ((𝑧𝑅 −s 𝐴) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
Theorem	precsexlem1 28623	Lemma for surreal reciprocals. Calculate the value of the recursive left function at zero. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ (𝐿‘∅) = { 0s }
Theorem	precsexlem2 28624	Lemma for surreal reciprocals. Calculate the value of the recursive right function at zero. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ (𝑅‘∅) = ∅
Theorem	precsexlem3 28625*	Lemma for surreal reciprocals. Calculate the value of the recursive function at a successor. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ (𝐼 ∈ ω → (𝐹‘suc 𝐼) = ⟨((𝐿‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), ((𝑅‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩)
Theorem	precsexlem4 28626*	Lemma for surreal reciprocals. Calculate the value of the recursive left function at a successor. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ (𝐼 ∈ ω → (𝐿‘suc 𝐼) = ((𝐿‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
Theorem	precsexlem5 28627*	Lemma for surreal reciprocals. Calculate the value of the recursive right function at a successor. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ (𝐼 ∈ ω → (𝑅‘suc 𝐼) = ((𝑅‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
Theorem	precsexlem6 28628*	Lemma for surreal reciprocal. Show that 𝐿 is non-strictly increasing in its argument. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘𝐽))
Theorem	precsexlem7 28629*	Lemma for surreal reciprocal. Show that 𝑅 is non-strictly increasing in its argument. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    ⇒   ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))
Theorem	precsexlem8 28630*	Lemma for surreal reciprocal. Show that the left and right functions give sets of surreals. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ ((L‘𝐴) ∪ (R‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   ⊢ ((𝜑 ∧ 𝐼 ∈ ω) → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No))
Theorem	precsexlem9 28631*	Lemma for surreal reciprocal. Show that the product of 𝐴 and a left element is less than one and the product of 𝐴 and a right element is greater than one. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ ((L‘𝐴) ∪ (R‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   ⊢ ((𝜑 ∧ 𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))
Theorem	precsexlem10 28632*	Lemma for surreal reciprocal. Show that the union of the left sets is less than the union of the right sets. Note that this is the first theorem in the surreal numbers to require the axiom of infinity. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ ((L‘𝐴) ∪ (R‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))
Theorem	precsexlem11 28633*	Lemma for surreal reciprocal. Show that the cut of the left and right sets is a multiplicative inverse for 𝐴. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ (L‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 −s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ (R‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 −s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   ⊢ 𝐿 = (1st ∘ 𝐹)    &   ⊢ 𝑅 = (2nd ∘ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ ((L‘𝐴) ∪ (R‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    &   ⊢ 𝑌 = (∪ (𝐿 “ ω) |s ∪ (𝑅 “ ω))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝑌) = 1s )
Theorem	precsex 28634*	Every positive surreal has a reciprocal. Theorem 10(iv) of [Conway] p. 21. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
Theorem	recsex 28635*	A nonzero surreal has a reciprocal. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
Theorem	recsexd 28636*	A nonzero surreal has a reciprocal. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐴 ≠ 0s )    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
Theorem	divmuls 28637	Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
Theorem	divmulsd 28638	Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
Theorem	divscl 28639	Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) ∈ No)
Theorem	divscld 28640	Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    ⇒   ⊢ (𝜑 → (𝐴 /su 𝐵) ∈ No)
Theorem	divscan2d 28641	A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    ⇒   ⊢ (𝜑 → (𝐵 ·s (𝐴 /su 𝐵)) = 𝐴)
Theorem	divscan1d 28642	A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐵) ·s 𝐵) = 𝐴)
Theorem	ltdivmulsd 28643	Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐶 ·s 𝐵)))
Theorem	ltdivmuls2d 28644	Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 ·s 𝐶)))
Theorem	ltmuldivsd 28645	Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
Theorem	ltmuldivs2d 28646	Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
Theorem	divsassd 28647	An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶)))
Theorem	divmuldivsd 28648	Multiplication of two surreal ratios. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    &   ⊢ (𝜑 → 𝐷 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐵) ·s (𝐶 /su 𝐷)) = ((𝐴 ·s 𝐶) /su (𝐵 ·s 𝐷)))
Theorem	divdivs1d 28649	Surreal division into a fraction. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 /su 𝐵) /su 𝐶) = (𝐴 /su (𝐵 ·s 𝐶)))
Theorem	divsrecd 28650	Relationship between surreal division and reciprocal. (Contributed by Scott Fenton, 13-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    ⇒   ⊢ (𝜑 → (𝐴 /su 𝐵) = (𝐴 ·s ( 1s /su 𝐵)))
Theorem	divsdird 28651	Distribution of surreal division over addition. (Contributed by Scott Fenton, 13-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 𝐶) = ((𝐴 /su 𝐶) +s (𝐵 /su 𝐶)))
Theorem	divscan3d 28652	A cancellation law for surreal division. (Contributed by Scott Fenton, 13-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐵 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐵 ·s 𝐴) /su 𝐵) = 𝐴)
15.5.5  Absolute value
Syntax	cabss 28653	Declare the syntax for surreal absolute value.
class abss
Definition	df-abss 28654	Define the surreal absolute value function. See abssval 28655 for its value and absscl 28656 for its closure. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ abss = (𝑥 ∈ No ↦ if( 0s ≤s 𝑥, 𝑥, ( -s‘𝑥)))
Theorem	abssval 28655	The value of surreal absolute value. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → (abss‘𝐴) = if( 0s ≤s 𝐴, 𝐴, ( -s‘𝐴)))
Theorem	absscl 28656	Closure law for surreal absolute value. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → (abss‘𝐴) ∈ No)
Theorem	abssid 28657	The absolute value of a non-negative surreal is itself. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
Theorem	abs0s 28658	The absolute value of surreal zero. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (abss‘ 0s ) = 0s
Theorem	abssnid 28659	For a negative surreal, its absolute value is its negation. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -s‘𝐴))
Theorem	absmuls 28660	Surreal absolute value distributes over multiplication. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
Theorem	abssge0 28661	The absolute value of a surreal number is non-negative. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → 0s ≤s (abss‘𝐴))
Theorem	abssor 28662	The absolute value of a surreal is either that surreal or its negative. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → ((abss‘𝐴) = 𝐴 ∨ (abss‘𝐴) = ( -s‘𝐴)))
Theorem	absnegs 28663	Surreal absolute value of the negative. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → (abss‘( -s‘𝐴)) = (abss‘𝐴))
Theorem	leabss 28664	A surreal is less than or equal to its absolute value. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝐴 ∈ No → 𝐴 ≤s (abss‘𝐴))
Theorem	abslts 28665	Surreal absolute value and less-than relation. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((abss‘𝐴) <s 𝐵 ↔ (( -s‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵)))
Theorem	abssubs 28666	Swapping order of surreal subtraction doesn't change the absolute value. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (abss‘(𝐴 −s 𝐵)) = (abss‘(𝐵 −s 𝐴)))
15.6  Subsystems of surreals
15.6.1  Ordinal numbers
Syntax	cons 28667	Declare the syntax for surreal ordinals.
class Ons
Definition	df-ons 28668	Define the surreal ordinals. These are the maximum members of each generation of surreals. Variant of definition from [Conway] p. 27. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ Ons = {𝑥 ∈ No ∣ (R‘𝑥) = ∅}
Theorem	elons 28669	Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ (R‘𝐴) = ∅))
Theorem	onssno 28670	The surreal ordinals are a subclass of the surreals. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ Ons ⊆ No
Theorem	onno 28671	A surreal ordinal is a surreal. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)
Theorem	0ons 28672	Surreal zero is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ 0s ∈ Ons
Theorem	1ons 28673	Surreal one is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
⊢ 1s ∈ Ons
Theorem	elons2 28674*	A surreal is ordinal iff it is the cut of some set of surreals and the empty set. Definition from [Conway] p. 27. (Contributed by Scott Fenton, 19-Mar-2025.)
⊢ (𝐴 ∈ Ons ↔ ∃𝑎 ∈ 𝒫 No𝐴 = (𝑎 |s ∅))
Theorem	elons2d 28675	The cut of any set of surreals and the empty set is a surreal ordinal. (Contributed by Scott Fenton, 19-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ No)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s ∅))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Ons)
Theorem	onleft 28676	The left set of a surreal ordinal is the same as its old set. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ (𝐴 ∈ Ons → (O‘(bday‘𝐴)) = (L‘𝐴))
Theorem	ltonold 28677*	The class of ordinals less than any surreal is a subset of that surreal's old set. (Contributed by Scott Fenton, 22-Mar-2025.)
⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ (O‘(bday‘𝐴)))
Theorem	ltonsex 28678*	The class of ordinals less than any particular surreal is a set. Theorem 14 of [Conway] p. 27. (Contributed by Scott Fenton, 22-Mar-2025.)
⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
Theorem	oncutleft 28679	A surreal ordinal is equal to the cut of its left set and the empty set. (Contributed by Scott Fenton, 29-Mar-2025.)
⊢ (𝐴 ∈ Ons → 𝐴 = ((L‘𝐴) |s ∅))
Theorem	oncutlt 28680*	A surreal ordinal is the simplest number greater than all previous surreal ordinals. Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 4-Nov-2025.)
⊢ (𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))
Theorem	bday11on 28681	The birthday function is one-to-one over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐴 = 𝐵)
Theorem	onnolt 28682	If a surreal ordinal is less than a given surreal, then it is simpler. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → (bday‘𝐴) ∈ (bday‘𝐵))
Theorem	onlts 28683	Less-than is the same as birthday comparison over surreal ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 <s 𝐵 ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
Theorem	onles 28684	Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ≤s 𝐵 ↔ (bday‘𝐴) ⊆ (bday‘𝐵)))
Theorem	onltsd 28685	Less-than is the same as birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ Ons)    &   ⊢ (𝜑 → 𝐵 ∈ Ons)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
Theorem	onlesd 28686	Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ Ons)    &   ⊢ (𝜑 → 𝐵 ∈ Ons)    ⇒   ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (bday‘𝐴) ⊆ (bday‘𝐵)))
Theorem	oniso 28687	The birthday function restricted to the surreal ordinals forms an order-preserving isomorphism with the regular ordinals. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ (bday ↾ Ons) Isom <s , E (Ons, On)
Theorem	onswe 28688	Surreal less-than well-orders the surreal ordinals. Part of Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ <s We Ons
Theorem	onsse 28689	Surreal less-than is set-like over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ <s Se Ons
Theorem	onsis 28690*	Transfinite induction schema for surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ Ons → (∀𝑦 ∈ Ons (𝑦 <s 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝐴 ∈ Ons → 𝜒)
Theorem	ons2ind 28691*	Double induction schema for surreal ordinals. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝑥 = 𝑥𝑂 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑦𝑂 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑥𝑂 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → 𝜒) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → 𝜓) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → 𝜃)) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → 𝜂)
Theorem	bdayons 28692*	The birthday of a surreal ordinal is the set of all previous ordinal birthdays. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ Ons → (bday‘𝐴) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))
Theorem	onaddscl 28693	The surreal ordinals are closed under addition. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)
Theorem	onmulscl 28694	The surreal ordinals are closed under multiplication. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)
Theorem	addonbday 28695	The birthday of the sum of two ordinals is the natural sum of their birthdays. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (bday‘(𝐴 +s 𝐵)) = ((bday‘𝐴) +no (bday‘𝐵)))
Theorem	peano2ons 28696	The successor of a surreal ordinal is a surreal ordinal. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ (𝐴 ∈ Ons → (𝐴 +s 1s ) ∈ Ons)
Theorem	onsbnd 28697	The surreals of a given birthday are bounded above by that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → 𝐵 ≤s 𝐴)
Theorem	onsbnd2 28698	The surreals of a given birthday are bounded below by the negative of that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → ( -s‘𝐴) ≤s 𝐵)
15.6.2  Surreal recursive sequences
Syntax	cseqs 28699	Extend class notation with the surreal recursive sequence builder.
class seqs𝑀( + , 𝐹)
Definition	df-seqs 28700*	Define a general-purpose sequence builder for surreal numbers. Compare df-seq 14172. Note that in the theorems we develop here, we do not require 𝑀 to be an integer. This is because there are infinite surreal numbers and we may want to start our sequence there. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)