P. 285 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	addsubsassd 28401	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 28402	Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 +s 𝐵) -s 𝐶) = ((𝐴 -s 𝐶) +s 𝐵))
Theorem	ltsubsubsbd 28403	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐷 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐶) <s (𝐵 -s 𝐷) ↔ (𝐴 -s 𝐵) <s (𝐶 -s 𝐷)))
Theorem	ltsubsubs2bd 28404	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	ltsubsubs3bd 28405	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	lesubsubsbd 28406	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	lesubsubs2bd 28407	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	lesubsubs3bd 28408	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	ltsubaddsd 28409	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	ltsubadds2d 28410	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	ltaddsubsd 28411	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	ltaddsubs2d 28412	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	lesubaddsd 28413	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 28414	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 28415	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	lesubsd 28416	Swap subtrahends in a surreal inequality. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 ≤s (𝐵 -s 𝐶) ↔ 𝐶 ≤s (𝐵 -s 𝐴)))
Theorem	nncansd 28417	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 28418	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	ltsubsposd 28419	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 28420	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 28421	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	ltsm1d 28422	A surreal is greater than itself minus one. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 -s 1s ) <s 𝐴)
Theorem	subscan1d 28423	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐶 -s 𝐴) = (𝐶 -s 𝐵) ↔ 𝐴 = 𝐵))
Theorem	subscan2d 28424	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐶) = (𝐵 -s 𝐶) ↔ 𝐴 = 𝐵))
Theorem	subseq0d 28425	The difference between two surreals is zero iff they are equal. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ((𝐴 -s 𝐵) = 0s ↔ 𝐴 = 𝐵))
15.5.3  Multiplication
Syntax	cmuls 28426	Set up the syntax for surreal multiplication.
class ·s
Definition	df-muls 28427*	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 28428	Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ·s Fn ( No × No )
Theorem	mulsval 28429*	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 28430*	Lemma for mulsval2 28431. 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 28431*	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 28432	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
Theorem	mulsrid 28433	Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)
Theorem	mulsridd 28434	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 28435*	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 28436*	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 28437*	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 28438*	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 28439*	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 28440*	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 28441*	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 28442*	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 28443*	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 28444*	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 28445*	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 28446*	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 28447*	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 28448*	Lemma for surreal multiplication. Remove the restriction on 𝐶 and 𝐷 from mulsproplem12 28447. (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 28449*	Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 28447 and mulsproplem13 28448. This completes the induction on surreal multiplication. mulsprop 28450 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 28450	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	mulcutlem 28451*	Lemma for mulcut 28452. 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	mulcut 28452*	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	mulcut2 28453*	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 28454	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 28455	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	ltmuls 28456	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	ltmulsd 28457	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	lemulsd 28458	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 28459	Surreal multiplication is commutative. 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 28460	Surreal multiplication is commutative. 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 28461	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0s ·s 𝐴) = 0s )
Theorem	mulslid 28462	Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
Theorem	mulslidd 28463	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 28464	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 28465	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 28466	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	sltmuls1 28467*	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	sltmuls2 28468*	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 28469*	Lemma for mulsunif 28470. 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 28470*	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 28471*	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 28472*	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 28473*	Lemma for addsdi 28475. 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 28474*	Lemma for addsdi 28475. 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 28475	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 28476	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 28477	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 28478	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 28479	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 28480	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 28481	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 28482	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 28483*	Lemma for mulsass 28486. 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 28484*	Lemma for mulsass 28486. 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 28485*	Lemma for mulsass 28486. 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 28486	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 28459, addsdi 28475, mulsgt0 28464, 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 28487	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 28488	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 28489*	Lemma for mulsunif2 28490. 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 28490*	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	ltmuls2 28491	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	ltmuls2d 28492	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	ltmuls1d 28493	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	lemuls2d 28494	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)))
Theorem	lemuls1d 28495	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 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
Theorem	ltmulnegs1d 28496	Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐶 <s 0s )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐵 ·s 𝐶) <s (𝐴 ·s 𝐶)))
Theorem	ltmulnegs2d 28497	Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐶 <s 0s )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 ·s 𝐵) <s (𝐶 ·s 𝐴)))
Theorem	mulscan2dlem 28498	Lemma for mulscan2d 28499. Cancellation of multiplication when the right term is positive. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
Theorem	mulscan2d 28499	Cancellation of surreal multiplication when the right term is nonzero. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
Theorem	mulscan1d 28500	Cancellation of surreal multiplication when the left term is nonzero. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐶 ≠ 0s )    ⇒   ⊢ (𝜑 → ((𝐶 ·s 𝐴) = (𝐶 ·s 𝐵) ↔ 𝐴 = 𝐵))