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	sleadd1 27901	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	sleadd2 27902	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	sltadd2 27903	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	sltadd1 27904	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 27905	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 27906	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	sleadd1d 27907	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	sleadd2d 27908	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	sltadd2d 27909	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	sltadd1d 27910	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 27911	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 27912	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 27913*	Lemma for addsunif 27914. 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 27914*	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 27915*	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 27916*	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 27917	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 27918	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 27919	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 27920	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 27921	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 27922	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	sltaddpos1d 27923	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	sltaddpos2d 27924	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	slt2addd 27925	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 27926	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	sltp1d 27927	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	addsbdaylem 27928*	Lemma for addsbday 27929. (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	addsbday 27929	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 27930	Declare the syntax for surreal negation.
class -us
Syntax	csubs 27931	Declare the syntax for surreal subtraction.
class -s
Definition	df-negs 27932*	Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
Definition	df-subs 27933*	Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
Theorem	negsfn 27934	Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -us Fn No
Theorem	subsfn 27935	Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ -s Fn ( No × No )
Theorem	negsval 27936	The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
Theorem	negs0s 27937	Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( -us ‘ 0s ) = 0s
Theorem	negs1s 27938	An expression for negative surreal one. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ ( -us ‘ 1s ) = (∅ |s { 0s })
Theorem	negsproplem1 27939*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))    ⇒   ⊢ (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))
Theorem	negsproplem2 27940*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Theorem	negsproplem3 27941*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
Theorem	negsproplem4 27942*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))    ⇒   ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Theorem	negsproplem5 27943*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))    ⇒   ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Theorem	negsproplem6 27944*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵))    ⇒   ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Theorem	negsproplem7 27945*	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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    ⇒   ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Theorem	negsprop 27946	Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
Theorem	negscl 27947	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
Theorem	negscld 27948	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → ( -us ‘𝐴) ∈ No )
Theorem	sltnegim 27949	The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))
Theorem	negscut 27950	The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
Theorem	negscut2 27951	The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Theorem	negsid 27952	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 ( -us ‘𝐴)) = 0s )
Theorem	negsidd 27953	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 ( -us ‘𝐴)) = 0s )
Theorem	negsex 27954*	Every surreal has a negative. Note that this theorem, addscl 27893, addscom 27878, addsass 27917, addsrid 27876, and sltadd1im 27897 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 27955	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 → ( -us ‘( -us ‘𝐴)) = 𝐴)
Theorem	sltneg 27956	Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
Theorem	sleneg 27957	Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴)))
Theorem	sltnegd 27958	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 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
Theorem	slenegd 27959	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 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴)))
Theorem	negs11 27960	Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) = ( -us ‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	negsdi 27961	Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))
Theorem	slt0neg2d 27962	Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘𝐴) <s 0s ))
Theorem	negsf 27963	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -us : No ⟶ No
Theorem	negsfo 27964	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -us : No –onto→ No
Theorem	negsf1o 27965	Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -us : No –1-1-onto→ No
Theorem	negsunif 27966	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 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))
Theorem	negsbdaylem 27967	Lemma for negsbday 27968. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))
Theorem	negsbday 27968	Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
Theorem	subsval 27969	The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
Theorem	subsvald 27970	The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
Theorem	subscl 27971	Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) ∈ No )
Theorem	subscld 27972	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 27973	Function statement for surreal subtraction. (Contributed by Scott Fenton, 17-May-2025.)
⊢ -s :( No × No )⟶ No
Theorem	subsfo 27974	Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.)
⊢ -s :( No × No )–onto→ No
Theorem	negsval2 27975	Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ No → ( -us ‘𝐴) = ( 0s -s 𝐴))
Theorem	negsval2d 27976	Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → ( -us ‘𝐴) = ( 0s -s 𝐴))
Theorem	subsid1 27977	Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 -s 0s ) = 𝐴)
Theorem	subsid 27978	Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 -s 𝐴) = 0s )
Theorem	subadds 27979	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 27980	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 27981	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐵) = 𝐴)
Theorem	pncan3s 27982	Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s (𝐵 -s 𝐴)) = 𝐵)
Theorem	pncan2s 27983	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐴) = 𝐵)
Theorem	npcans 27984	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴)
Theorem	sltsub1 27985	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	sltsub2 27986	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	sltsub1d 27987	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	sltsub2d 27988	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 27989	Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ( -us ‘(𝐴 -s 𝐵)) = (𝐵 -s 𝐴))
Theorem	addsubsassd 27990	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 27991	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	sltsubsubbd 27992	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	sltsubsub2bd 27993	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)))
Theorem	sltsubsub3bd 27994	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 𝐶) <s (𝐵 -s 𝐷) ↔ (𝐷 -s 𝐶) <s (𝐵 -s 𝐴)))
Theorem	slesubsubbd 27995	Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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	slesubsub2bd 27996	Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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)))
Theorem	slesubsub3bd 27997	Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐶) ≤s (𝐵 -s 𝐷) ↔ (𝐷 -s 𝐶) ≤s (𝐵 -s 𝐴)))
Theorem	sltsubaddd 27998	Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐶 +s 𝐵)))
Theorem	sltsubadd2d 27999	Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐵 +s 𝐶)))
Theorem	sltaddsubd 28000	Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐶 -s 𝐵)))