P. 285 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltadds1im 28401	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	ltadds2im 28402	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	leadds1im 28403	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	leadds2im 28404	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	leadds1 28405	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)))
Theorem	leadds2 28406	Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (𝐴 ≤s 𝐵 ↔ (𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵)))
Theorem	ltadds2 28407	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵)))
Theorem	ltadds1 28408	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (𝐴 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐵 +s 𝐶)))
Theorem	addscan2 28409	Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐴 +s 𝐶) = (𝐵 +s 𝐶) ↔ 𝐴 = 𝐵))
Theorem	addscan1 28410	Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐶 +s 𝐴) = (𝐶 +s 𝐵) ↔ 𝐴 = 𝐵))
Theorem	leadds1d 28411	Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
Theorem	leadds2d 28412	Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵)))
Theorem	ltadds2d 28413	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵)))
Theorem	ltadds1d 28414	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐵 +s 𝐶)))
Theorem	addscan2d 28415	Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐶) = (𝐵 +s 𝐶) ↔ 𝐴 = 𝐵))
Theorem	addscan1d 28416	Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐶 +s 𝐴) = (𝐶 +s 𝐵) ↔ 𝐴 = 𝐵))
Theorem	addsuniflem 28417*	Lemma for addsunif 28418. State the whole theorem with extra distinct variable conditions. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})))
Theorem	addsunif 28418*	Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define 𝐴 and 𝐵 in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})))
Theorem	addsasslem1 28419*	Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ (L‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ (L‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s (({𝑎 ∣ ∃𝑝 ∈ (R‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ (R‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ (R‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))
Theorem	addsasslem2 28420*	Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ (L‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ (R‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ (R‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ (R‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})))
Theorem	addsass 28421	Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))
Theorem	addsassd 28422	Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶)))
Theorem	adds32d 28423	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((𝐴 +s 𝐶) +s 𝐵))
Theorem	adds12d 28424	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (𝐵 +s (𝐴 +s 𝐶)))
Theorem	adds4d 28425	Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) +s (𝐶 +s 𝐷)) = ((𝐴 +s 𝐶) +s (𝐵 +s 𝐷)))
Theorem	adds42d 28426	Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) +s (𝐶 +s 𝐷)) = ((𝐴 +s 𝐶) +s (𝐷 +s 𝐵)))
Theorem	ltaddspos1d 28427	Addition of a positive number increases the sum. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ( 0s <s 𝐴 ↔ 𝐵 <s (𝐵 +s 𝐴)))
Theorem	ltaddspos2d 28428	Addition of a positive number increases the sum. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ( 0s <s 𝐴 ↔ 𝐵 <s (𝐴 +s 𝐵)))
Theorem	lt2addsd 28429	Adding both sides of two surreal less-than relations. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    &   ⊢ (𝜑 → 𝐴 <s 𝐶)    &   ⊢ (𝜑 → 𝐵 <s 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 +s 𝐵) <s (𝐶 +s 𝐷))
Theorem	addsgt0d 28430	The sum of two positive surreals is positive. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → 0s <s 𝐵)    ⇒   ⊢ (𝜑 → 0s <s (𝐴 +s 𝐵))
Theorem	ltsp1d 28431	A surreal is less than itself plus one. (Contributed by Scott Fenton, 13-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
Theorem	addsge01d 28432	A surreal is less-than or equal to itself plus a non-negative surreal. (Contributed by Scott Fenton, 24-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ( 0s ≤s 𝐵 ↔ 𝐴 ≤s (𝐴 +s 𝐵)))
Theorem	addbdaylem 28433*	Lemma for addbday 28434. (Contributed by Scott Fenton, 13-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → ∀𝑦𝑂 ∈ ((L‘𝐵) ∪ (R‘𝐵))(bday‘(𝐴 +s 𝑦𝑂)) ⊆ ((bday‘𝐴) +no (bday‘𝑦𝑂)))    &   ⊢ 𝑆 ⊆ ((L‘𝐵) ∪ (R‘𝐵))    ⇒   ⊢ (𝜑 → (bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ ((bday‘𝐴) +no (bday‘𝐵)))
Theorem	addbday 28434	The birthday of the sum of two surreals is less than or equal to the natural ordinal sum of their individual birthdays. Theorem 6.1 of [Gonshor] p. 95. (Contributed by Scott Fenton, 12-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (bday‘(𝐴 +s 𝐵)) ⊆ ((bday‘𝐴) +no (bday‘𝐵)))
15.5.2  Negation and Subtraction
Syntax	cnegs 28435	Declare the syntax for surreal negation.
class -s
Syntax	csubs 28436	Declare the syntax for surreal subtraction.
class −s
Definition	df-negs 28437*	Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥)))))
Definition	df-subs 28438*	Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ −s = (𝑥 ∈ No, 𝑦 ∈ No ↦ (𝑥 +s ( -s‘𝑦)))
Theorem	negsfn 28439	Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -s Fn No
Theorem	subsfn 28440	Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ −s Fn (No × No)
Theorem	negsval 28441	The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ (𝐴 ∈ No → ( -s‘𝐴) = (( -s “ (R‘𝐴)) |s ( -s “ (L‘𝐴))))
Theorem	neg0s 28442	Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( -s‘ 0s ) = 0s
Theorem	neg1s 28443	An expression for negative surreal one. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ ( -s‘ 1s ) = (∅ |s { 0s })
Theorem	negsproplem1 28444*	Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No)    &   ⊢ (𝜑 → 𝑌 ∈ No)    &   ⊢ (𝜑 → ((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))    ⇒   ⊢ (𝜑 → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))
Theorem	negsproplem2 28445*	Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → ( -s “ (R‘𝐴)) <<s ( -s “ (L‘𝐴)))
Theorem	negsproplem3 28446*	Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → (( -s‘𝐴) ∈ No ∧ ( -s “ (R‘𝐴)) <<s {( -s‘𝐴)} ∧ {( -s‘𝐴)} <<s ( -s “ (L‘𝐴))))
Theorem	negsproplem4 28447*	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → (bday‘𝐴) ∈ (bday‘𝐵))    ⇒   ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))
Theorem	negsproplem5 28448*	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → (bday‘𝐵) ∈ (bday‘𝐴))    ⇒   ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))
Theorem	negsproplem6 28449*	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is the same age as 𝐵. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → (bday‘𝐴) = (bday‘𝐵))    ⇒   ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))
Theorem	negsproplem7 28450*	Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    ⇒   ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))
Theorem	negsprop 28451	Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴))))
Theorem	negscl 28452	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
Theorem	negscld 28453	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → ( -s‘𝐴) ∈ No)
Theorem	ltnegsim 28454	The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 <s 𝐵 → ( -s‘𝐵) <s ( -s‘𝐴)))
Theorem	negcut 28455	The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (( -s‘𝐴) ∈ No ∧ ( -s “ (R‘𝐴)) <<s {( -s‘𝐴)} ∧ {( -s‘𝐴)} <<s ( -s “ (L‘𝐴))))
Theorem	negcut2 28456	The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -s “ (R‘𝐴)) <<s ( -s “ (L‘𝐴)))
Theorem	negsid 28457	Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 +s ( -s‘𝐴)) = 0s )
Theorem	negsidd 28458	Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 +s ( -s‘𝐴)) = 0s )
Theorem	negsex 28459*	Every surreal has a negative. Note that this theorem, addscl 28397, addscom 28382, addsass 28421, addsrid 28380, and ltadds1im 28401 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s )
Theorem	negnegs 28460	A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -s‘( -s‘𝐴)) = 𝐴)
Theorem	ltnegs 28461	Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 <s 𝐵 ↔ ( -s‘𝐵) <s ( -s‘𝐴)))
Theorem	lenegs 28462	Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ≤s 𝐵 ↔ ( -s‘𝐵) ≤s ( -s‘𝐴)))
Theorem	ltnegsd 28463	Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ( -s‘𝐵) <s ( -s‘𝐴)))
Theorem	lenegsd 28464	Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ ( -s‘𝐵) ≤s ( -s‘𝐴)))
Theorem	negs11 28465	Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (( -s‘𝐴) = ( -s‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	negsdi 28466	Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ( -s‘(𝐴 +s 𝐵)) = (( -s‘𝐴) +s ( -s‘𝐵)))
Theorem	lt0negs2d 28467	Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → ( 0s <s 𝐴 ↔ ( -s‘𝐴) <s 0s ))
Theorem	negsf 28468	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -s:No⟶No
Theorem	negsfo 28469	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -s:No⟶ontoNo
Theorem	negsf1o 28470	Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -s:No⟶1-1ontoNo
Theorem	negsunif 28471	Uniformity property for surreal negation. If 𝐿 and 𝑅 are any cut that represents 𝐴, then they may be used instead of (L‘𝐴) and (R‘𝐴) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    ⇒   ⊢ (𝜑 → ( -s‘𝐴) = (( -s “ 𝑅) |s ( -s “ 𝐿)))
Theorem	negbdaylem 28472	Lemma for negbday 28473. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) ⊆ (bday‘𝐴))
Theorem	negbday 28473	Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) = (bday‘𝐴))
Theorem	negleft 28474	The left set of the negative of a surreal is the set of negatives of its right set. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ No → (L‘( -s‘𝐴)) = ( -s “ (R‘𝐴)))
Theorem	negright 28475	The right set of the negative of a surreal is the set of negatives of its left set. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ No → (R‘( -s‘𝐴)) = ( -s “ (L‘𝐴)))
Theorem	subsval 28476	The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 −s 𝐵) = (𝐴 +s ( -s‘𝐵)))
Theorem	subsvald 28477	The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 −s 𝐵) = (𝐴 +s ( -s‘𝐵)))
Theorem	subscl 28478	Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 −s 𝐵) ∈ No)
Theorem	subscld 28479	Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 −s 𝐵) ∈ No)
Theorem	subsf 28480	Function statement for surreal subtraction. (Contributed by Scott Fenton, 17-May-2025.)
⊢ −s :(No × No)⟶No
Theorem	subsfo 28481	Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.)
⊢ −s :(No × No)⟶ontoNo
Theorem	negsval2 28482	Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ No → ( -s‘𝐴) = ( 0s −s 𝐴))
Theorem	negsval2d 28483	Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    ⇒   ⊢ (𝜑 → ( -s‘𝐴) = ( 0s −s 𝐴))
Theorem	subsid1 28484	Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 −s 0s ) = 𝐴)
Theorem	subsid 28485	Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 −s 𝐴) = 0s )
Theorem	subadds 28486	Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → ((𝐴 −s 𝐵) = 𝐶 ↔ (𝐵 +s 𝐶) = 𝐴))
Theorem	subaddsd 28487	Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 −s 𝐵) = 𝐶 ↔ (𝐵 +s 𝐶) = 𝐴))
Theorem	pncans 28488	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((𝐴 +s 𝐵) −s 𝐵) = 𝐴)
Theorem	pncan3s 28489	Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s (𝐵 −s 𝐴)) = 𝐵)
Theorem	pncan2s 28490	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((𝐴 +s 𝐵) −s 𝐴) = 𝐵)
Theorem	npcans 28491	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ((𝐴 −s 𝐵) +s 𝐵) = 𝐴)
Theorem	ltsubs1 28492	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (𝐴 <s 𝐵 ↔ (𝐴 −s 𝐶) <s (𝐵 −s 𝐶)))
Theorem	ltsubs2 28493	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (𝐴 <s 𝐵 ↔ (𝐶 −s 𝐵) <s (𝐶 −s 𝐴)))
Theorem	ltsubs1d 28494	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 −s 𝐶) <s (𝐵 −s 𝐶)))
Theorem	ltsubs2d 28495	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 −s 𝐵) <s (𝐶 −s 𝐴)))
Theorem	negsubsdi2d 28496	Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    ⇒   ⊢ (𝜑 → ( -s‘(𝐴 −s 𝐵)) = (𝐵 −s 𝐴))
Theorem	addsubsassd 28497	Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) −s 𝐶) = (𝐴 +s (𝐵 −s 𝐶)))
Theorem	addsubsd 28498	Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) −s 𝐶) = ((𝐴 −s 𝐶) +s 𝐵))
Theorem	ltsubsubsbd 28499	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 −s 𝐶) <s (𝐵 −s 𝐷) ↔ (𝐴 −s 𝐵) <s (𝐶 −s 𝐷)))
Theorem	ltsubsubs2bd 28500	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No)    &   ⊢ (𝜑 → 𝐵 ∈ No)    &   ⊢ (𝜑 → 𝐶 ∈ No)    &   ⊢ (𝜑 → 𝐷 ∈ No)    ⇒   ⊢ (𝜑 → ((𝐴 −s 𝐵) <s (𝐶 −s 𝐷) ↔ (𝐷 −s 𝐶) <s (𝐵 −s 𝐴)))