P. 283 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cutminmax 28201*	If the left set of 𝑋 has a maximum and the right set of 𝑋 has a minimum, then 𝑋 is equal to the cut of the maximum and the minimum. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))    &   ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝑋))    &   ⊢ (𝜑 → ∀𝑦 ∈ ( R ‘𝑋)𝑅 ≤s 𝑦)    ⇒   ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))
15.4  Induction and recursion
15.4.1  Induction and recursion on one variable
Syntax	cnorec 28202	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
Definition	df-norec 28203*	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 28204*	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 28205*	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 28206*	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 28207*	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 28208*	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 28209*	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 28210*	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 28211	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 28212	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 28213	Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
Definition	df-norec2 28214*	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 28215*	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 28216*	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 28217*	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 28218*	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	no2indlesm 28219*	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 28220*	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 28221	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 28222	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 28223*	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 28224	Declare the syntax for surreal addition.
class +s
Definition	df-adds 28225*	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 28226	Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ +s Fn ( No × No )
Theorem	addsval 28227*	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 28228*	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 28229	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 28230	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 28231	Surreal addition is commutative. 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 28232	Surreal addition is commutative. 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))
Theorem	addslid 28233	Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴)
Theorem	addsproplem1 28234*	Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8388 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (( bday ‘𝐴) +no ( bday ‘𝐶))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Theorem	addsproplem2 28235*	Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))
Theorem	addsproplem3 28236*	Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addsproplem4 28237*	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem5 28238*	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem6 28239*	Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    &   ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsproplem7 28240*	Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑌 <s 𝑍)    ⇒   ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Theorem	addsprop 28241	Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))
Theorem	addcutslem 28242*	Lemma for addcuts 28243. Show the statement with some additional distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addcuts 28243*	Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
Theorem	addcuts2 28244*	Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))
Theorem	addscld 28245	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    ⇒   ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
Theorem	addscl 28246	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
Theorem	addsf 28247	Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +s :( No × No )⟶ No
Theorem	addsfo 28248	Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +s :( No × No )–onto→ No
Theorem	peano2no 28249	A theorem for surreals that is analogous to the second Peano postulate peano2 7886. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (𝐴 ∈ No → (𝐴 +s 1s ) ∈ No )
Theorem	ltadds1im 28250	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 28251	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 28252	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 28253	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 28254	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 28255	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 28256	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 28257	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 28258	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 28259	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 28260	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 28261	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 28262	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 28263	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 28264	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 28265	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 28266*	Lemma for addsunif 28267. 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 28267*	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 28268*	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 28269*	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 28270	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 28271	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 28272	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 28273	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 28274	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 28275	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 28276	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 28277	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 28278	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 28279	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 28280	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 28281	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 28282*	Lemma for addbday 28283. (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 28283	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 28284	Declare the syntax for surreal negation.
class -us
Syntax	csubs 28285	Declare the syntax for surreal subtraction.
class -s
Definition	df-negs 28286*	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 28287*	Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
Theorem	negsfn 28288	Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -us Fn No
Theorem	subsfn 28289	Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ -s Fn ( No × No )
Theorem	negsval 28290	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	neg0s 28291	Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( -us ‘ 0s ) = 0s
Theorem	neg1s 28292	An expression for negative surreal one. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ ( -us ‘ 1s ) = (∅ |s { 0s })
Theorem	negsproplem1 28293*	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 28294*	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 28295*	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 28296*	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 28297*	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 28298*	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 28299*	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 28300	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))))