P. 282 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	subaddsd 28101	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 28102	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐵) = 𝐴)
Theorem	pncan3s 28103	Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s (𝐵 -s 𝐴)) = 𝐵)
Theorem	pncan2s 28104	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐴) = 𝐵)
Theorem	npcans 28105	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴)
Theorem	sltsub1 28106	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 28107	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 28108	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 28109	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 28110	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 28111	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 28112	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 28113	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 28114	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 28115	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 28116	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 28117	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 28118	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 28119	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 28120	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 28121	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)))
Theorem	sltaddsub2d 28122	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 ↔ 𝐵 <s (𝐶 -s 𝐴)))
Theorem	slesubaddd 28123	Surreal less-than or equal relationship between subtraction and addition. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) ≤s 𝐶 ↔ 𝐴 ≤s (𝐶 +s 𝐵)))
Theorem	subsubs4d 28124	Law for double surreal subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) -s 𝐶) = (𝐴 -s (𝐵 +s 𝐶)))
Theorem	subsubs2d 28125	Law for double surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 -s (𝐵 -s 𝐶)) = (𝐴 +s (𝐶 -s 𝐵)))
Theorem	nncansd 28126	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 -s (𝐴 -s 𝐵)) = 𝐵)
Theorem	posdifsd 28127	Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐴)))
Theorem	sltsubposd 28128	Subtraction of a positive number decreases 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	subsge0d 28129	Non-negative subtraction. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ( 0s ≤s (𝐴 -s 𝐵) ↔ 𝐵 ≤s 𝐴))
Theorem	addsubs4d 28130	Rearrangement of four terms in mixed addition and subtraction. Surreal version. (Contributed by Scott Fenton, 25-Jul-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	sltm1d 28131	A surreal is greater than itself minus one. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 -s 1s ) <s 𝐴)
15.5.3  Multiplication
Syntax	cmuls 28132	Set up the syntax for surreal multiplication.
class ·s
Definition	df-muls 28133*	Define surreal multiplication. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ·s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +s (𝑥𝑚𝑞)) -s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +s (𝑥𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +s (𝑥𝑚𝑢)) -s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +s (𝑥𝑚𝑤)) -s (𝑣𝑚𝑤))}))))
Theorem	mulsfn 28134	Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ·s Fn ( No × No )
Theorem	mulsval 28135*	The value of surreal multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	mulsval2lem 28136*	Lemma for mulsval2 28137. Change bound variables in one of the cases. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ {𝑎 ∣ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑏 ∣ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}
Theorem	mulsval2 28137*	The value of surreal multiplication, expressed with fewer distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	muls01 28138	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
Theorem	mulsrid 28139	Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)
Theorem	mulsridd 28140	Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s 1s ) = 𝐴)
Theorem	mulsproplemcbv 28141*	Lemma for surreal multiplication. Change some bound variables for later use. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    ⇒   ⊢ (𝜑 → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
Theorem	mulsproplem1 28142*	Lemma for surreal multiplication. Instantiate some variables. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝑋 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ No )    &   ⊢ (𝜑 → 𝑍 ∈ No )    &   ⊢ (𝜑 → 𝑊 ∈ No )    &   ⊢ (𝜑 → 𝑇 ∈ No )    &   ⊢ (𝜑 → 𝑈 ∈ No )    &   ⊢ (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘𝑍) +no ( bday ‘𝑇)) ∪ (( bday ‘𝑊) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑍) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑊) +no ( bday ‘𝑇))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))    ⇒   ⊢ (𝜑 → ((𝑋 ·s 𝑌) ∈ No ∧ ((𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈) → ((𝑍 ·s 𝑈) -s (𝑍 ·s 𝑇)) <s ((𝑊 ·s 𝑈) -s (𝑊 ·s 𝑇)))))
Theorem	mulsproplem2 28143*	Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and 𝐵 itself is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝑋 ·s 𝐵) ∈ No )
Theorem	mulsproplem3 28144*	Lemma for surreal multiplication. Under the inductive hypothesis, the product of 𝐴 itself and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝑌) ∈ No )
Theorem	mulsproplem4 28145*	Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))    &   ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))    ⇒   ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )
Theorem	mulsproplem5 28146*	Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))    &   ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵))    ⇒   ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
Theorem	mulsproplem6 28147*	Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))    &   ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))    &   ⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴))    &   ⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵))    ⇒   ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
Theorem	mulsproplem7 28148*	Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝐴))    &   ⊢ (𝜑 → 𝑆 ∈ ( R ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵))    ⇒   ⊢ (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)))
Theorem	mulsproplem8 28149*	Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝐴))    &   ⊢ (𝜑 → 𝑆 ∈ ( R ‘𝐵))    &   ⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴))    &   ⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵))    ⇒   ⊢ (𝜑 → (((𝑅 ·s 𝐵) +s (𝐴 ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
Theorem	mulsproplem9 28150*	Lemma for surreal multiplication. Show that the cut involved in surreal multiplication makes sense. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
Theorem	mulsproplem10 28151*	Lemma for surreal multiplication. State the cut properties of surreal multiplication. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	mulsproplem11 28152*	Lemma for surreal multiplication. Under the inductive hypothesis, demonstrate closure of surreal multiplication. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
Theorem	mulsproplem12 28153*	Lemma for surreal multiplication. Demonstrate the second half of the inductive statement assuming 𝐶 and 𝐷 are not the same age and 𝐸 and 𝐹 are not the same age. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    &   ⊢ (𝜑 → 𝐸 ∈ No )    &   ⊢ (𝜑 → 𝐹 ∈ No )    &   ⊢ (𝜑 → 𝐶 <s 𝐷)    &   ⊢ (𝜑 → 𝐸 <s 𝐹)    &   ⊢ (𝜑 → (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)))    &   ⊢ (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
Theorem	mulsproplem13 28154*	Lemma for surreal multiplication. Remove the restriction on 𝐶 and 𝐷 from mulsproplem12 28153. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    &   ⊢ (𝜑 → 𝐸 ∈ No )    &   ⊢ (𝜑 → 𝐹 ∈ No )    &   ⊢ (𝜑 → 𝐶 <s 𝐷)    &   ⊢ (𝜑 → 𝐸 <s 𝐹)    &   ⊢ (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
Theorem	mulsproplem14 28155*	Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 28153 and mulsproplem13 28154. This completes the induction on surreal multiplication. mulsprop 28156 brings all this together technically. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    &   ⊢ (𝜑 → 𝐸 ∈ No )    &   ⊢ (𝜑 → 𝐹 ∈ No )    &   ⊢ (𝜑 → 𝐶 <s 𝐷)    &   ⊢ (𝜑 → 𝐸 <s 𝐹)    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
Theorem	mulsprop 28156	Surreals are closed under multiplication and obey a particular ordering law. Theorem 3.4 of [Gonshor] p. 17. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))))
Theorem	mulscutlem 28157*	Lemma for mulscut 28158. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	mulscut 28158*	Show the cut properties of surreal multiplication. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	mulscut2 28159*	Show that the cut involved in surreal multiplication is actually a cut. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
Theorem	mulscl 28160	The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
Theorem	mulscld 28161	The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
Theorem	sltmul 28162	An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No )) → ((𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) <s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
Theorem	sltmuld 28163	An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → 𝐶 <s 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) <s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
Theorem	slemuld 28164	An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≤s 𝐵)    &   ⊢ (𝜑 → 𝐶 ≤s 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
Theorem	mulscom 28165	Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
Theorem	mulscomd 28166	Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
Theorem	muls02 28167	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0s ·s 𝐴) = 0s )
Theorem	mulslid 28168	Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
Theorem	mulslidd 28169	Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → ( 1s ·s 𝐴) = 𝐴)
Theorem	mulsgt0 28170	The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))
Theorem	mulsgt0d 28171	The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐴)    &   ⊢ (𝜑 → 0s <s 𝐵)    ⇒   ⊢ (𝜑 → 0s <s (𝐴 ·s 𝐵))
Theorem	mulsge0d 28172	The product of two non-negative surreals is non-negative. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 0s ≤s 𝐴)    &   ⊢ (𝜑 → 0s ≤s 𝐵)    ⇒   ⊢ (𝜑 → 0s ≤s (𝐴 ·s 𝐵))
Theorem	ssltmul1 28173*	One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)})
Theorem	ssltmul2 28174*	One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
Theorem	mulsuniflem 28175*	Lemma for mulsunif 28176. State the theorem with some extra distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	mulsunif 28176*	Surreal multiplication has the uniformity property. That is, any cuts that define 𝐴 and 𝐵 can be used in the definition of (𝐴 ·s 𝐵). Theorem 3.5 of [Gonshor] p. 18. (Contributed by Scott Fenton, 7-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Theorem	addsdilem1 28177*	Lemma for surreal distribution. Expand the left hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))))
Theorem	addsdilem2 28178*	Lemma for surreal distribution. Expand the right hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))))
Theorem	addsdilem3 28179*	Lemma for addsdi 28181. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)))    &   ⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))    &   ⊢ (𝜓 → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) -s (𝑋 ·s (𝑌 +s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))
Theorem	addsdilem4 28180*	Lemma for addsdi 28181. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))    &   ⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))    &   ⊢ (𝜓 → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
Theorem	addsdi 28181	Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
Theorem	addsdid 28182	Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
Theorem	addsdird 28183	Distributive law for surreal numbers. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) ·s 𝐶) = ((𝐴 ·s 𝐶) +s (𝐵 ·s 𝐶)))
Theorem	subsdid 28184	Distribution of surreal multiplication over subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s (𝐵 -s 𝐶)) = ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶)))
Theorem	subsdird 28185	Distribution of surreal multiplication over subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) ·s 𝐶) = ((𝐴 ·s 𝐶) -s (𝐵 ·s 𝐶)))
Theorem	mulnegs1d 28186	Product with negative is negative of product. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
Theorem	mulnegs2d 28187	Product with negative is negative of product. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
Theorem	mul2negsd 28188	Surreal product of two negatives. (Contributed by Scott Fenton, 15-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (( -us ‘𝐴) ·s ( -us ‘𝐵)) = (𝐴 ·s 𝐵))
Theorem	mulsasslem1 28189*	Lemma for mulsass 28192. Expand the left hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))))
Theorem	mulsasslem2 28190*	Lemma for mulsass 28192. Expand the right hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}))))
Theorem	mulsasslem3 28191*	Lemma for mulsass 28192. Demonstrate the central equality. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ 𝑃 ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))    &   ⊢ 𝑄 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))    &   ⊢ 𝑅 ⊆ (( L ‘𝐶) ∪ ( R ‘𝐶))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝑥𝑂 ·s 𝑦𝑂) ·s 𝐶) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s 𝐵) ·s 𝑧𝑂) = (𝑥𝑂 ·s (𝐵 ·s 𝑧𝑂)))    &   ⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝑦𝑂) ·s 𝑧𝑂) = (𝐴 ·s (𝑦𝑂 ·s 𝑧𝑂)))    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))((𝑥𝑂 ·s 𝐵) ·s 𝐶) = (𝑥𝑂 ·s (𝐵 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝐴 ·s 𝑦𝑂) ·s 𝐶) = (𝐴 ·s (𝑦𝑂 ·s 𝐶)))    &   ⊢ (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝐵) ·s 𝑧𝑂) = (𝐴 ·s (𝐵 ·s 𝑧𝑂)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))))))
Theorem	mulsass 28192	Associative law for surreal multiplication. Part of theorem 7 of [Conway] p. 19. Much like the case for additive groups, this theorem together with mulscom 28165, addsdi 28181, mulsgt0 28170, and the addition theorems would make the surreals into an ordered ring except that they are a proper class. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))
Theorem	mulsassd 28193	Associative law for surreal multiplication. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))
Theorem	muls4d 28194	Rearrangement of four surreal factors. (Contributed by Scott Fenton, 16-Apr-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	mulsunif2lem 28195*	Lemma for mulsunif2 28196. State the theorem with extra disjoint variable conditions. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})))
Theorem	mulsunif2 28196*	Alternate expression for surreal multiplication. Note from [Conway] p. 19. (Contributed by Scott Fenton, 16-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})))
Theorem	sltmul2 28197	Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))
Theorem	sltmul2d 28198	Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 ·s 𝐴) <s (𝐶 ·s 𝐵)))
Theorem	sltmul1d 28199	Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ·s 𝐶) <s (𝐵 ·s 𝐶)))
Theorem	slemul2d 28200	Multiplication of both sides of surreal less-than or equal by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐶 ·s 𝐴) ≤s (𝐶 ·s 𝐵)))