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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addsf 27701 | Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ +s :( No × No )⟶ No | ||
Theorem | addsfo 27702 | Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ +s :( No × No )–onto→ No | ||
Theorem | peano2no 27703 | A theorem for surreals that is analagous to the second Peano postulate peano2 7884. (Contributed by Scott Fenton, 17-Mar-2025.) |
⊢ (𝐴 ∈ No → (𝐴 +s 1s ) ∈ No ) | ||
Theorem | sltadd1im 27704 | Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐴 +s 𝐶) <s (𝐵 +s 𝐶))) | ||
Theorem | sltadd2im 27705 | Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐶 +s 𝐴) <s (𝐶 +s 𝐵))) | ||
Theorem | sleadd1im 27706 | Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵)) | ||
Theorem | sleadd2im 27707 | Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵) → 𝐴 ≤s 𝐵)) | ||
Theorem | sleadd1 27708 | Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶))) | ||
Theorem | sleadd2 27709 | Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵))) | ||
Theorem | sltadd2 27710 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵))) | ||
Theorem | sltadd1 27711 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐵 +s 𝐶))) | ||
Theorem | addscan2 27712 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) = (𝐵 +s 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | addscan1 27713 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +s 𝐴) = (𝐶 +s 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | sleadd1d 27714 | Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶))) | ||
Theorem | sleadd2d 27715 | Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵))) | ||
Theorem | sltadd2d 27716 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵))) | ||
Theorem | sltadd1d 27717 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐵 +s 𝐶))) | ||
Theorem | addscan2d 27718 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐶) = (𝐵 +s 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | addscan1d 27719 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐶 +s 𝐴) = (𝐶 +s 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | addsuniflem 27720* | Lemma for addsunif 27721. State the whole theorem with extra distinct variable conditions. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝑀 <<s 𝑆) & ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) & ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) ⇒ ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) | ||
Theorem | addsunif 27721* | Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define 𝐴 and 𝐵 in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝑀 <<s 𝑆) & ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) & ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) ⇒ ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) | ||
Theorem | addsasslem1 27722* | Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)}))) | ||
Theorem | addsasslem2 27723* | Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))) | ||
Theorem | addsass 27724 | Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶))) | ||
Theorem | addsassd 27725 | Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶))) | ||
Theorem | adds32d 27726 | Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((𝐴 +s 𝐶) +s 𝐵)) | ||
Theorem | adds12d 27727 | Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (𝐵 +s (𝐴 +s 𝐶))) | ||
Theorem | adds4d 27728 | Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) +s (𝐶 +s 𝐷)) = ((𝐴 +s 𝐶) +s (𝐵 +s 𝐷))) | ||
Theorem | adds42d 27729 | Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) +s (𝐶 +s 𝐷)) = ((𝐴 +s 𝐶) +s (𝐷 +s 𝐵))) | ||
Syntax | cnegs 27730 | Declare the syntax for surreal negation. |
class -us | ||
Syntax | csubs 27731 | Declare the syntax for surreal subtraction. |
class -s | ||
Definition | df-negs 27732* | Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))) | ||
Definition | df-subs 27733* | Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦))) | ||
Theorem | negsfn 27734 | Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ -us Fn No | ||
Theorem | subsfn 27735 | Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.) |
⊢ -s Fn ( No × No ) | ||
Theorem | negsval 27736 | The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))) | ||
Theorem | negs0s 27737 | Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ ( -us ‘ 0s ) = 0s | ||
Theorem | negsproplem1 27738* | Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵))) ⇒ ⊢ (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋)))) | ||
Theorem | negsproplem2 27739* | Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴))) | ||
Theorem | negsproplem3 27740* | Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)))) | ||
Theorem | negsproplem4 27741* | Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (Contributed by Scott Fenton, 2-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)) ⇒ ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | ||
Theorem | negsproplem5 27742* | Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) ⇒ ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | ||
Theorem | negsproplem6 27743* | Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is the same age as 𝐵. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵)) ⇒ ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | ||
Theorem | negsproplem7 27744* | Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) ⇒ ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | ||
Theorem | negsprop 27745 | Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))) | ||
Theorem | negscl 27746 | The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No ) | ||
Theorem | negscld 27747 | The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → ( -us ‘𝐴) ∈ No ) | ||
Theorem | sltnegim 27748 | The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))) | ||
Theorem | negscut 27749 | The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)))) | ||
Theorem | negscut2 27750 | The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴))) | ||
Theorem | negsid 27751 | Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s ) | ||
Theorem | negsidd 27752 | Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +s ( -us ‘𝐴)) = 0s ) | ||
Theorem | negsex 27753* | Every surreal has a negative. Note that this theorem, addscl 27700, addscom 27685, addsass 27724, addsrid 27683, and sltadd1im 27704 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.) |
⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +s 𝑥) = 0s ) | ||
Theorem | negnegs 27754 | A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴) | ||
Theorem | sltneg 27755 | Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴))) | ||
Theorem | sleneg 27756 | Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴))) | ||
Theorem | sltnegd 27757 | Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴))) | ||
Theorem | slenegd 27758 | Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴))) | ||
Theorem | negs11 27759 | Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) = ( -us ‘𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | negsdi 27760 | Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵))) | ||
Theorem | slt0neg2d 27761 | Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘𝐴) <s 0s )) | ||
Theorem | negsf 27762 | Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ -us : No ⟶ No | ||
Theorem | negsfo 27763 | Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ -us : No –onto→ No | ||
Theorem | negsf1o 27764 | Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ -us : No –1-1-onto→ No | ||
Theorem | negsunif 27765 | Uniformity property for surreal negation. If 𝐿 and 𝑅 are any cut that represents 𝐴, then they may be used instead of ( L ‘𝐴) and ( R ‘𝐴) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.) |
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) ⇒ ⊢ (𝜑 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿))) | ||
Theorem | negsbdaylem 27766 | Lemma for negsbday 27767. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.) |
⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴)) | ||
Theorem | negsbday 27767 | Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.) |
⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴)) | ||
Theorem | subsval 27768 | The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) | ||
Theorem | subsvald 27769 | The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵))) | ||
Theorem | subscl 27770 | Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) ∈ No ) | ||
Theorem | subscld 27771 | Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 -s 𝐵) ∈ No ) | ||
Theorem | subsid1 27772 | Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → (𝐴 -s 0s ) = 𝐴) | ||
Theorem | subsid 27773 | Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → (𝐴 -s 𝐴) = 0s ) | ||
Theorem | subadds 27774 | Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -s 𝐵) = 𝐶 ↔ (𝐵 +s 𝐶) = 𝐴)) | ||
Theorem | subaddsd 27775 | 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 27776 | Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐵) = 𝐴) | ||
Theorem | pncan3s 27777 | Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s (𝐵 -s 𝐴)) = 𝐵) | ||
Theorem | npcans 27778 | Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴) | ||
Theorem | sltsub1 27779 | 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 27780 | 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 27781 | 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 27782 | 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 27783 | 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 27784 | 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 27785 | 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 27786 | 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 27787 | 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 27788 | 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 27789 | 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 27790 | 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 27791 | 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 27792 | 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 27793 | 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 27794 | 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 27795 | 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 | subsubs4d 27796 | 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 | posdifsd 27797 | 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))) | ||
Syntax | cmuls 27798 | Set up the syntax for surreal multiplication. |
class ·s | ||
Definition | df-muls 27799* | 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 27800 | Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ·s Fn ( No × No ) |
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