P. 341 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34001-34100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rightirr 34001	No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( R ‘𝑋)
Theorem	left0s 34002	The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( L ‘ 0s ) = ∅
Theorem	right0s 34003	The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( R ‘ 0s ) = ∅
Theorem	lrold 34004	The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
Theorem	madebdaylemold 34005*	Lemma for madebday 34007. If the inductive hypothesis of madebday 34007 is satisfied, the converse of oldbdayim 33998 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
Theorem	madebdaylemlrcut 34006*	Lemma for madebday 34007. If the inductive hypothesis of madebday 34007 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 34010 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	madebday 34007	A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
Theorem	oldbday 34008	A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
Theorem	newbday 34009	A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))
Theorem	lrcut 34010	A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	scutfo 34011	The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ |s : <<s –onto→ No
Theorem	sltn0 34012	If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))
Theorem	lruneq 34013	If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
Theorem	sltlpss 34014	If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
20.9.32  Surreal numbers: Cofinality and coinitiality
Theorem	cofsslt 34015*	If every element of 𝐴 is bounded by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Theorem	coinitsslt 34016*	If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
Theorem	cofcut1 34017*	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷. Then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut2 34018*	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcutr 34019*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧))
Theorem	cofcutrtime 34020*	If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
20.9.33  Surreal numbers: Induction and recursion on one variable
Syntax	cnorec 34021	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)
Definition	df-norec 34022*	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 34023*	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 34024*	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 34025*	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 34026*	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 34027*	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 34028*	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 34029*	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 34030	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 34031	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)))))
20.9.34  Surreal numbers: Induction and recursion on two variables
Syntax	cnorec2 34032	Declare the syntax for surreal recursion on two arguments.
class norec2 (𝐹)
Definition	df-norec2 34033*	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 34034*	To get through most of the textbook defintions 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 34035*	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 34036*	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 34037*	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	no2indslem 34038*	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))}    &   ⊢ 𝑆 = {⟨𝑐, 𝑑⟩ ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))}    &   ⊢ (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 34039*	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 34040	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 34041	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 34042*	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)
20.9.35  Surreal numbers - addition, negation, and subtraction
Syntax	cadds 34043	Declare the syntax for surreal addition.
class +s
Syntax	cnegs 34044	Declare the syntax for surreal negation.
class -us
Syntax	csubs 34045	Declare the syntax for surreal subtraction.
class -s
Definition	df-adds 34046*	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)}))))
Definition	df-negs 34047*	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 34048*	Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
Theorem	negsfn 34049	Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ -us Fn No
Theorem	negsval 34050	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 34051	Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( -us ‘ 0s ) = 0s
Theorem	addsfn 34052	Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ +s Fn ( No × No )
Theorem	addsval 34053*	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	addsid1 34054	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	addsid1d 34055	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 34056	Surreal addition commutes. 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 34057	Surreal addition commutes. 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	addscllem1 34058	Lemma for addscl (future) Alternate expression for surreal addition. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))))
20.9.36  Quantifier-free definitions
Syntax	ctxp 34059	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 34060	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 34061	Declare the subset relationship class.
class SSet
Syntax	ctrans 34062	Declare the transitive set class.
class Trans
Syntax	cbigcup 34063	Declare the set union relationship.
class Bigcup
Syntax	cfix 34064	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 34065	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 34066	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 34067	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 34068	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 34069	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 34070	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 34071	Declare the syntax for the image function.
class Img
Syntax	cdomain 34072	Declare the syntax for the domain function.
class Domain
Syntax	crange 34073	Declare the syntax for the range function.
class Range
Syntax	capply 34074	Declare the syntax for the application function.
class Apply
Syntax	ccup 34075	Declare the syntax for the cup function.
class Cup
Syntax	ccap 34076	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 34077	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 34078	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 34079	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 34080	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 34081	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 34082	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 34083	Define the tail cross of two classes. Membership in this class is defined by txpss3v 34107 and brtxp 34109. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 34084	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 34113 and brpprod 34114. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 34085	Define the subset class. For the value, see brsset 34118. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 34086	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 34087	Define the Bigcup function, which, per fvbigcup 34131, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
Definition	df-fix 34088	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 34089	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 34090	Define the class of all functions. See elfuns 34144 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 34091	Define the singleton function. See brsingle 34146 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 34092	Define the class of all singletons. See elsingles 34147 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 34093	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 34159 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 34094	Define the cartesian product function. See brcart 34161 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 34095	Define the image function. See brimg 34166 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 34096	Define the domain function. See brdomain 34162 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 34097	Define the range function. See brrange 34163 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 34098	Define the little cup function. See brcup 34168 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V)))
Definition	df-cap 34099	Define the little cap function. See brcap 34169 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗ V)))
Definition	df-restrict 34100	Define the restriction function. See brrestrict 34178 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))