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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegs0s 27501 Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
( -us ‘ 0s ) = 0s
 
Theoremnegsproplem1 27502* Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → (( bday ‘𝑋) ∊ ( bday ‘𝑌)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)))    ⇒   (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))
 
Theoremnegsproplem2 27503* Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    ⇒   (𝜑 → ( -us “ ( R ‘ðī)) <<s ( -us “ ( L ‘ðī)))
 
Theoremnegsproplem3 27504* Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    ⇒   (𝜑 → (( -us ‘ðī) ∈ No ∧ ( -us “ ( R ‘ðī)) <<s {( -us ‘ðī)} ∧ {( -us ‘ðī)} <<s ( -us “ ( L ‘ðī))))
 
Theoremnegsproplem4 27505* Lemma for surreal negation. Show the second half of the inductive hypothesis when ðī is simpler than ðĩ. (Contributed by Scott Fenton, 2-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðī <s ðĩ)    &   (𝜑 → ( bday ‘ðī) ∈ ( bday ‘ðĩ))    ⇒   (𝜑 → ( -us ‘ðĩ) <s ( -us ‘ðī))
 
Theoremnegsproplem5 27506* Lemma for surreal negation. Show the second half of the inductive hypothesis when ðĩ is simpler than ðī. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðī <s ðĩ)    &   (𝜑 → ( bday ‘ðĩ) ∈ ( bday ‘ðī))    ⇒   (𝜑 → ( -us ‘ðĩ) <s ( -us ‘ðī))
 
Theoremnegsproplem6 27507* Lemma for surreal negation. Show the second half of the inductive hypothesis when ðī is the same age as ðĩ. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðī <s ðĩ)    &   (𝜑 → ( bday ‘ðī) = ( bday ‘ðĩ))    ⇒   (𝜑 → ( -us ‘ðĩ) <s ( -us ‘ðī))
 
Theoremnegsproplem7 27508* Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → ∀ð‘Ĩ ∈ No ∀ð‘Ķ ∈ No ((( bday ‘ð‘Ĩ) ∊ ( bday ‘ð‘Ķ)) ∈ (( bday ‘ðī) ∊ ( bday ‘ðĩ)) → (( -us ‘ð‘Ĩ) ∈ No ∧ (ð‘Ĩ <s ð‘Ķ → ( -us ‘ð‘Ķ) <s ( -us ‘ð‘Ĩ)))))    &   (𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðī <s ðĩ)    ⇒   (𝜑 → ( -us ‘ðĩ) <s ( -us ‘ðī))
 
Theoremnegsprop 27509 Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (( -us ‘ðī) ∈ No ∧ (ðī <s ðĩ → ( -us ‘ðĩ) <s ( -us ‘ðī))))
 
Theoremnegscl 27510 The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → ( -us ‘ðī) ∈ No )
 
Theoremnegscld 27511 The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → ðī ∈ No )    ⇒   (𝜑 → ( -us ‘ðī) ∈ No )
 
Theoremsltnegim 27512 The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī <s ðĩ → ( -us ‘ðĩ) <s ( -us ‘ðī)))
 
Theoremnegscut 27513 The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → (( -us ‘ðī) ∈ No ∧ ( -us “ ( R ‘ðī)) <<s {( -us ‘ðī)} ∧ {( -us ‘ðī)} <<s ( -us “ ( L ‘ðī))))
 
Theoremnegscut2 27514 The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → ( -us “ ( R ‘ðī)) <<s ( -us “ ( L ‘ðī)))
 
Theoremnegsid 27515 Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → (ðī +s ( -us ‘ðī)) = 0s )
 
Theoremnegsidd 27516 Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    ⇒   (𝜑 → (ðī +s ( -us ‘ðī)) = 0s )
 
Theoremnegsex 27517* Every surreal has a negative. Note that this theorem, addscl 27465, addscom 27450, addsass 27488, addsrid 27448, and sltadd1im 27468 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → ∃ð‘Ĩ ∈ No (ðī +s ð‘Ĩ) = 0s )
 
Theoremnegnegs 27518 A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → ( -us ‘( -us ‘ðī)) = ðī)
 
Theoremsltneg 27519 Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī <s ðĩ ↔ ( -us ‘ðĩ) <s ( -us ‘ðī)))
 
Theoremsleneg 27520 Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī â‰Īs ðĩ ↔ ( -us ‘ðĩ) â‰Īs ( -us ‘ðī)))
 
Theoremsltnegd 27521 Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī <s ðĩ ↔ ( -us ‘ðĩ) <s ( -us ‘ðī)))
 
Theoremslenegd 27522 Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī â‰Īs ðĩ ↔ ( -us ‘ðĩ) â‰Īs ( -us ‘ðī)))
 
Theoremnegs11 27523 Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (( -us ‘ðī) = ( -us ‘ðĩ) ↔ ðī = ðĩ))
 
Theoremnegsdi 27524 Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → ( -us ‘(ðī +s ðĩ)) = (( -us ‘ðī) +s ( -us ‘ðĩ)))
 
Theoremslt0neg2d 27525 Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → ðī ∈ No )    ⇒   (𝜑 → ( 0s <s ðī ↔ ( -us ‘ðī) <s 0s ))
 
Theoremnegsf 27526 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No âŸķ No
 
Theoremnegsfo 27527 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –onto→ No
 
Theoremnegsf1o 27528 Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –1-1-onto→ No
 
Theoremnegsunif 27529 Uniformity property for surreal negation. If ðŋ and 𝑅 are any cut that represents ðī, then they may be used instead of ( L ‘ðī) and ( R ‘ðī) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.)
(𝜑 → ðŋ <<s 𝑅)    &   (𝜑 → ðī = (ðŋ |s 𝑅))    ⇒   (𝜑 → ( -us ‘ðī) = (( -us “ 𝑅) |s ( -us “ ðŋ)))
 
Theoremnegsbdaylem 27530 Lemma for negsbday 27531. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)
(ðī ∈ No → ( bday ‘( -us ‘ðī)) ⊆ ( bday ‘ðī))
 
Theoremnegsbday 27531 Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.)
(ðī ∈ No → ( bday ‘( -us ‘ðī)) = ( bday ‘ðī))
 
Theoremsubsval 27532 The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī -s ðĩ) = (ðī +s ( -us ‘ðĩ)))
 
Theoremsubsvald 27533 The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī -s ðĩ) = (ðī +s ( -us ‘ðĩ)))
 
Theoremsubscl 27534 Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī -s ðĩ) ∈ No )
 
Theoremsubscld 27535 Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī -s ðĩ) ∈ No )
 
Theoremsubsid1 27536 Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → (ðī -s 0s ) = ðī)
 
Theoremsubsid 27537 Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
(ðī ∈ No → (ðī -s ðī) = 0s )
 
Theoremsubadds 27538 Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → ((ðī -s ðĩ) = ðķ ↔ (ðĩ +s ðķ) = ðī))
 
Theoremsubaddsd 27539 Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) = ðķ ↔ (ðĩ +s ðķ) = ðī))
 
Theorempncans 27540 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → ((ðī +s ðĩ) -s ðĩ) = ðī)
 
Theorempncan3s 27541 Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī +s (ðĩ -s ðī)) = ðĩ)
 
Theoremnpcans 27542 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → ((ðī -s ðĩ) +s ðĩ) = ðī)
 
Theoremsltsub1 27543 Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī <s ðĩ ↔ (ðī -s ðķ) <s (ðĩ -s ðķ)))
 
Theoremsltsub2 27544 Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī <s ðĩ ↔ (ðķ -s ðĩ) <s (ðķ -s ðī)))
 
Theoremsltsub1d 27545 Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → (ðī <s ðĩ ↔ (ðī -s ðķ) <s (ðĩ -s ðķ)))
 
Theoremsltsub2d 27546 Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → (ðī <s ðĩ ↔ (ðķ -s ðĩ) <s (ðķ -s ðī)))
 
Theoremnegsubsdi2d 27547 Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ( -us ‘(ðī -s ðĩ)) = (ðĩ -s ðī))
 
Theoremaddsubsassd 27548 Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī +s ðĩ) -s ðķ) = (ðī +s (ðĩ -s ðķ)))
 
Theoremaddsubsd 27549 Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī +s ðĩ) -s ðķ) = ((ðī -s ðķ) +s ðĩ))
 
Theoremsltsubsubbd 27550 Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðķ) <s (ðĩ -s 𝐷) ↔ (ðī -s ðĩ) <s (ðķ -s 𝐷)))
 
Theoremsltsubsub2bd 27551 Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) <s (ðķ -s 𝐷) ↔ (𝐷 -s ðķ) <s (ðĩ -s ðī)))
 
Theoremsltsubsub3bd 27552 Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðķ) <s (ðĩ -s 𝐷) ↔ (𝐷 -s ðķ) <s (ðĩ -s ðī)))
 
Theoremslesubsubbd 27553 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðķ) â‰Īs (ðĩ -s 𝐷) ↔ (ðī -s ðĩ) â‰Īs (ðķ -s 𝐷)))
 
Theoremslesubsub2bd 27554 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) â‰Īs (ðķ -s 𝐷) ↔ (𝐷 -s ðķ) â‰Īs (ðĩ -s ðī)))
 
Theoremslesubsub3bd 27555 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((ðī -s ðķ) â‰Īs (ðĩ -s 𝐷) ↔ (𝐷 -s ðķ) â‰Īs (ðĩ -s ðī)))
 
Theoremsltsubaddd 27556 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) <s ðķ ↔ ðī <s (ðķ +s ðĩ)))
 
Theoremsltsubadd2d 27557 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) <s ðķ ↔ ðī <s (ðĩ +s ðķ)))
 
Theoremsltaddsubd 27558 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī +s ðĩ) <s ðķ ↔ ðī <s (ðķ -s ðĩ)))
 
Theoremsltaddsub2d 27559 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī +s ðĩ) <s ðķ ↔ ðĩ <s (ðķ -s ðī)))
 
Theoremsubsubs4d 27560 Law for double surreal subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    ⇒   (𝜑 → ((ðī -s ðĩ) -s ðķ) = (ðī -s (ðĩ +s ðķ)))
 
Theoremposdifsd 27561 Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī <s ðĩ ↔ 0s <s (ðĩ -s ðī)))
 
15.5.3  Multiplication
 
Syntaxcmuls 27562 Set up the syntax for surreal multiplication.
class ·s
 
Definitiondf-muls 27563* Define surreal multiplication. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 4-Feb-2025.)
·s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V â†Ķ âĶ‹(1st ‘𝑧) / ð‘ĨâĶŒâĶ‹(2nd ‘𝑧) / ð‘ĶâĶŒ(({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ð‘Ĩ)∃𝑞 ∈ ( L ‘ð‘Ķ)𝑎 = (((𝑝𝑚ð‘Ķ) +s (ð‘Ĩ𝑚𝑞)) -s (𝑝𝑚𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ð‘Ĩ)∃𝑠 ∈ ( R ‘ð‘Ķ)𝑏 = (((𝑟𝑚ð‘Ķ) +s (ð‘Ĩ𝑚𝑠)) -s (𝑟𝑚𝑠))}) |s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ð‘Ĩ)∃ð‘Ē ∈ ( R ‘ð‘Ķ)𝑐 = (((ð‘Ąð‘šð‘Ķ) +s (ð‘Ĩ𝑚ð‘Ē)) -s (ð‘Ąð‘šð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ð‘Ĩ)∃ð‘Ī ∈ ( L ‘ð‘Ķ)𝑑 = (((ð‘Ģ𝑚ð‘Ķ) +s (ð‘Ĩ𝑚ð‘Ī)) -s (ð‘Ģ𝑚ð‘Ī))}))))
 
Theoremmulsfn 27564 Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
·s Fn ( No × No )
 
Theoremmulsval 27565* The value of surreal multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī ·s ðĩ) = (({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑐 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑑 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))})))
 
Theoremmulsval2lem 27566* Lemma for mulsval2 27567. Change bound variables in one of the cases. (Contributed by Scott Fenton, 8-Mar-2025.)
{𝑎 âˆĢ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑏 âˆĢ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}
 
Theoremmulsval2 27567* The value of surreal multiplication, expressed with fewer distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī ·s ðĩ) = (({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑐 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑑 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))})))
 
Theoremmuls01 27568 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(ðī ∈ No → (ðī ·s 0s ) = 0s )
 
Theoremmulsrid 27569 Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(ðī ∈ No → (ðī ·s 1s ) = ðī)
 
Theoremmulsridd 27570 Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → ðī ∈ No )    ⇒   (𝜑 → (ðī ·s 1s ) = ðī)
 
Theoremmulsproplemcbv 27571* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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))))))
 
Theoremmulsproplem1 27572* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))))    ⇒   (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)))))
 
Theoremmulsproplem2 27573* Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of ðī and ðĩ 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 ‘ðī)))    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (𝑋 ·s ðĩ) ∈ No )
 
Theoremmulsproplem3 27574* Lemma for surreal multiplication. Under the inductive hypothesis, the product of ðī itself and a member of the old set of ðĩ 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → 𝑌 ∈ ( O ‘( bday ‘ðĩ)))    ⇒   (𝜑 → (ðī ·s 𝑌) ∈ No )
 
Theoremmulsproplem4 27575* Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of ðī and a member of the old set of ðĩ 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 ‘ðī)))    &   (𝜑 → 𝑌 ∈ ( O ‘( bday ‘ðĩ)))    ⇒   (𝜑 → (𝑋 ·s 𝑌) ∈ No )
 
Theoremmulsproplem5 27576* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → 𝑃 ∈ ( L ‘ðī))    &   (𝜑 → 𝑄 ∈ ( L ‘ðĩ))    &   (𝜑 → 𝑇 ∈ ( L ‘ðī))    &   (𝜑 → 𝑈 ∈ ( R ‘ðĩ))    ⇒   (𝜑 → (((𝑃 ·s ðĩ) +s (ðī ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑇 ·s ðĩ) +s (ðī ·s 𝑈)) -s (𝑇 ·s 𝑈)))
 
Theoremmulsproplem6 27577* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → 𝑃 ∈ ( L ‘ðī))    &   (𝜑 → 𝑄 ∈ ( L ‘ðĩ))    &   (𝜑 → 𝑉 ∈ ( R ‘ðī))    &   (𝜑 → 𝑊 ∈ ( L ‘ðĩ))    ⇒   (𝜑 → (((𝑃 ·s ðĩ) +s (ðī ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s ðĩ) +s (ðī ·s 𝑊)) -s (𝑉 ·s 𝑊)))
 
Theoremmulsproplem7 27578* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → 𝑅 ∈ ( R ‘ðī))    &   (𝜑 → 𝑆 ∈ ( R ‘ðĩ))    &   (𝜑 → 𝑇 ∈ ( L ‘ðī))    &   (𝜑 → 𝑈 ∈ ( R ‘ðĩ))    ⇒   (𝜑 → (((𝑅 ·s ðĩ) +s (ðī ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑇 ·s ðĩ) +s (ðī ·s 𝑈)) -s (𝑇 ·s 𝑈)))
 
Theoremmulsproplem8 27579* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → 𝑅 ∈ ( R ‘ðī))    &   (𝜑 → 𝑆 ∈ ( R ‘ðĩ))    &   (𝜑 → 𝑉 ∈ ( R ‘ðī))    &   (𝜑 → 𝑊 ∈ ( L ‘ðĩ))    ⇒   (𝜑 → (((𝑅 ·s ðĩ) +s (ðī ·s 𝑆)) -s (𝑅 ·s 𝑆)) <s (((𝑉 ·s ðĩ) +s (ðī ·s 𝑊)) -s (𝑉 ·s 𝑊)))
 
Theoremmulsproplem9 27580* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ({𝑔 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑔 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {ℎ âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)ℎ = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑖 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑖 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑗 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑗 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))}))
 
Theoremmulsproplem10 27581* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ((ðī ·s ðĩ) ∈ No ∧ ({𝑔 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑔 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {ℎ âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)ℎ = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(ðī ·s ðĩ)} ∧ {(ðī ·s ðĩ)} <<s ({𝑖 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑖 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑗 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑗 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))})))
 
Theoremmulsproplem11 27582* 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī ·s ðĩ) ∈ No )
 
Theoremmulsproplem12 27583* Lemma for surreal multiplication. Demonstrate the second half of the inductive statement assuming ðķ and 𝐷 are not the same age and ðļ and ðđ 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → ðļ ∈ No )    &   (𝜑 → ðđ ∈ No )    &   (𝜑 → ðķ <s 𝐷)    &   (𝜑 → ðļ <s ðđ)    &   (𝜑 → (( bday ‘ðķ) ∈ ( bday ‘𝐷) âˆĻ ( bday ‘𝐷) ∈ ( bday ‘ðķ)))    &   (𝜑 → (( bday ‘ðļ) ∈ ( bday ‘ðđ) âˆĻ ( bday ‘ðđ) ∈ ( bday ‘ðļ)))    ⇒   (𝜑 → ((ðķ ·s ðđ) -s (ðķ ·s ðļ)) <s ((𝐷 ·s ðđ) -s (𝐷 ·s ðļ)))
 
Theoremmulsproplem13 27584* Lemma for surreal multiplication. Remove the restriction on ðķ and 𝐷 from mulsproplem12 27583. (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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → ðļ ∈ No )    &   (𝜑 → ðđ ∈ No )    &   (𝜑 → ðķ <s 𝐷)    &   (𝜑 → ðļ <s ðđ)    &   (𝜑 → (( bday ‘ðļ) ∈ ( bday ‘ðđ) âˆĻ ( bday ‘ðđ) ∈ ( bday ‘ðļ)))    ⇒   (𝜑 → ((ðķ ·s ðđ) -s (ðķ ·s ðļ)) <s ((𝐷 ·s ðđ) -s (𝐷 ·s ðļ)))
 
Theoremmulsproplem14 27585* Lemma for surreal multiplication. Finally, we remove the restriction on ðļ and ðđ from mulsproplem12 27583 and mulsproplem13 27584. This completes the induction on surreal multiplication. mulsprop 27586 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 ‘ðī) +no ( bday ‘ðĩ)) ∊ (((( bday ‘ðķ) +no ( bday ‘ðļ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðđ))) ∊ ((( bday ‘ðķ) +no ( bday ‘ðđ)) ∊ (( bday ‘𝐷) +no ( bday ‘ðļ))))) → ((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 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → ðļ ∈ No )    &   (𝜑 → ðđ ∈ No )    &   (𝜑 → ðķ <s 𝐷)    &   (𝜑 → ðļ <s ðđ)    ⇒   (𝜑 → ((ðķ ·s ðđ) -s (ðķ ·s ðļ)) <s ((𝐷 ·s ðđ) -s (𝐷 ·s ðļ)))
 
Theoremmulsprop 27586 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.)
(((ðī ∈ No ∧ ðĩ ∈ No ) ∧ (ðķ ∈ No ∧ 𝐷 ∈ No ) ∧ (ðļ ∈ No ∧ ðđ ∈ No )) → ((ðī ·s ðĩ) ∈ No ∧ ((ðķ <s 𝐷 ∧ ðļ <s ðđ) → ((ðķ ·s ðđ) -s (ðķ ·s ðļ)) <s ((𝐷 ·s ðđ) -s (𝐷 ·s ðļ)))))
 
Theoremmulscutlem 27587* Lemma for mulscut 27588. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ((ðī ·s ðĩ) ∈ No ∧ ({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(ðī ·s ðĩ)} ∧ {(ðī ·s ðĩ)} <<s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑐 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑑 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))})))
 
Theoremmulscut 27588* Show the cut properties of surreal multiplication. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ((ðī ·s ðĩ) ∈ No ∧ ({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(ðī ·s ðĩ)} ∧ {(ðī ·s ðĩ)} <<s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑐 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑑 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))})))
 
Theoremmulscut2 27589* Show that the cut involved in surreal multiplication is actually a cut. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → ({𝑎 âˆĢ ∃𝑝 ∈ ( L ‘ðī)∃𝑞 ∈ ( L ‘ðĩ)𝑎 = (((𝑝 ·s ðĩ) +s (ðī ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∊ {𝑏 âˆĢ ∃𝑟 ∈ ( R ‘ðī)∃𝑠 ∈ ( R ‘ðĩ)𝑏 = (((𝑟 ·s ðĩ) +s (ðī ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑐 âˆĢ âˆƒð‘Ą ∈ ( L ‘ðī)∃ð‘Ē ∈ ( R ‘ðĩ)𝑐 = (((ð‘Ą ·s ðĩ) +s (ðī ·s ð‘Ē)) -s (ð‘Ą ·s ð‘Ē))} ∊ {𝑑 âˆĢ ∃ð‘Ģ ∈ ( R ‘ðī)∃ð‘Ī ∈ ( L ‘ðĩ)𝑑 = (((ð‘Ģ ·s ðĩ) +s (ðī ·s ð‘Ī)) -s (ð‘Ģ ·s ð‘Ī))}))
 
Theoremmulscl 27590 The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī ·s ðĩ) ∈ No )
 
Theoremmulscld 27591 The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī ·s ðĩ) ∈ No )
 
Theoremsltmul 27592 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
(((ðī ∈ No ∧ ðĩ ∈ No ) ∧ (ðķ ∈ No ∧ 𝐷 ∈ No )) → ((ðī <s ðĩ ∧ ðķ <s 𝐷) → ((ðī ·s 𝐷) -s (ðī ·s ðķ)) <s ((ðĩ ·s 𝐷) -s (ðĩ ·s ðķ))))
 
Theoremsltmuld 27593 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → ðī <s ðĩ)    &   (𝜑 → ðķ <s 𝐷)    ⇒   (𝜑 → ((ðī ·s 𝐷) -s (ðī ·s ðķ)) <s ((ðĩ ·s 𝐷) -s (ðĩ ·s ðķ)))
 
Theoremslemuld 27594 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    &   (𝜑 → ðķ ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → ðī â‰Īs ðĩ)    &   (𝜑 → ðķ â‰Īs 𝐷)    ⇒   (𝜑 → ((ðī ·s 𝐷) -s (ðī ·s ðķ)) â‰Īs ((ðĩ ·s 𝐷) -s (ðĩ ·s ðķ)))
 
Theoremmulscom 27595 Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
((ðī ∈ No ∧ ðĩ ∈ No ) → (ðī ·s ðĩ) = (ðĩ ·s ðī))
 
Theoremmulscomd 27596 Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → ðī ∈ No )    &   (𝜑 → ðĩ ∈ No )    ⇒   (𝜑 → (ðī ·s ðĩ) = (ðĩ ·s ðī))
 
Theoremmuls02 27597 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(ðī ∈ No → ( 0s ·s ðī) = 0s )
 
Theoremmulslid 27598 Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(ðī ∈ No → ( 1s ·s ðī) = ðī)
 
Theoremmulslidd 27599 Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → ðī ∈ No )    ⇒   (𝜑 → ( 1s ·s ðī) = ðī)
 
Theoremmulsgt0 27600 The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
(((ðī ∈ No ∧ 0s <s ðī) ∧ (ðĩ ∈ No ∧ 0s <s ðĩ)) → 0s <s (ðī ·s ðĩ))
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