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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elmade2 27601* | Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋))) | ||
Theorem | elold 27602* | Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) | ||
Theorem | ssltleft 27603 | A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴}) | ||
Theorem | ssltright 27604 | A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴)) | ||
Theorem | lltropt 27605 | The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.) (Revised by Scott Fenton, 21-Feb-2025.) |
⊢ ( L ‘𝐴) <<s ( R ‘𝐴) | ||
Theorem | made0 27606 | The only surreal made on day ∅ is 0s. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ ( M ‘∅) = { 0s } | ||
Theorem | new0 27607 | The only surreal new on day ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ ( N ‘∅) = { 0s } | ||
Theorem | old1 27608 | The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ( O ‘1o) = { 0s } | ||
Theorem | madess 27609 | If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵)) | ||
Theorem | oldssmade 27610 | The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) | ||
Theorem | leftssold 27611 | The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋)) | ||
Theorem | rightssold 27612 | The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋)) | ||
Theorem | leftssno 27613 | The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( L ‘𝐴) ⊆ No | ||
Theorem | rightssno 27614 | The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( R ‘𝐴) ⊆ No | ||
Theorem | madecut 27615 | Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴)) | ||
Theorem | madeun 27616 | The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)) | ||
Theorem | madeoldsuc 27617 | The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴)) | ||
Theorem | oldsuc 27618 | The value of the old set at a successor. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ (𝐴 ∈ On → ( O ‘suc 𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))) | ||
Theorem | oldlim 27619 | The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → ( O ‘𝐴) = ∪ ( O “ 𝐴)) | ||
Theorem | madebdayim 27620 | If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.) |
⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴) | ||
Theorem | oldbdayim 27621 | If 𝑋 is in the old set for 𝐴, then the birthday of 𝑋 is less than 𝐴. (Contributed by Scott Fenton, 10-Aug-2024.) |
⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴) | ||
Theorem | oldirr 27622 | No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.) |
⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋)) | ||
Theorem | leftirr 27623 | No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ¬ 𝑋 ∈ ( L ‘𝑋) | ||
Theorem | rightirr 27624 | No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ ¬ 𝑋 ∈ ( R ‘𝑋) | ||
Theorem | left0s 27625 | The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ ( L ‘ 0s ) = ∅ | ||
Theorem | right0s 27626 | The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ ( R ‘ 0s ) = ∅ | ||
Theorem | left1s 27627 | The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ( L ‘ 1s ) = { 0s } | ||
Theorem | right1s 27628 | The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.) |
⊢ ( R ‘ 1s ) = ∅ | ||
Theorem | lrold 27629 | The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴)) | ||
Theorem | madebdaylemold 27630* | Lemma for madebday 27632. If the inductive hypothesis of madebday 27632 is satisfied, the converse of oldbdayim 27621 holds. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴))) | ||
Theorem | madebdaylemlrcut 27631* | Lemma for madebday 27632. If the inductive hypothesis of madebday 27632 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27635 holds. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋) | ||
Theorem | madebday 27632 | A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴)) | ||
Theorem | oldbday 27633 | A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴)) | ||
Theorem | newbday 27634 | A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴)) | ||
Theorem | lrcut 27635 | A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋) | ||
Theorem | scutfo 27636 | The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.) |
⊢ |s : <<s –onto→ No | ||
Theorem | sltn0 27637 | If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.) |
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅)) | ||
Theorem | lruneq 27638 | If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.) |
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌))) | ||
Theorem | sltlpss 27639 | If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.) |
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌))) | ||
Theorem | slelss 27640 | If two surreals 𝐴 and 𝐵 share a birthday, then 𝐴 ≤s 𝐵 if and only if the left set of 𝐴 is a non-strict subset of the left set of 𝐵. (Contributed by Scott Fenton, 21-Mar-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵))) | ||
Theorem | 0elold 27641 | Zero is in the old set of any non-zero number. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐴 ≠ 0s ) ⇒ ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴))) | ||
Theorem | 0elleft 27642 | Zero is in the left set of any positive number. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 0s <s 𝐴) ⇒ ⊢ (𝜑 → 0s ∈ ( L ‘𝐴)) | ||
Theorem | 0elright 27643 | Zero is in the right set of any negative number. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 0s ) ⇒ ⊢ (𝜑 → 0s ∈ ( R ‘𝐴)) | ||
Theorem | cofsslt 27644* | If every element of 𝐴 is bounded above by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.) |
⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶) | ||
Theorem | coinitsslt 27645* | If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.) |
⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵) | ||
Theorem | cofcut1 27646* | If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.) |
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | ||
Theorem | cofcut1d 27647* | If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) & ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)}) & ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷) ⇒ ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | ||
Theorem | cofcut2 27648* | If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.) |
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | ||
Theorem | cofcut2d 27649* | If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) & ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) & ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) & ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) ⇒ ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | ||
Theorem | cofcutr 27650* | If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.) |
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧)) | ||
Theorem | cofcutr1d 27651* | If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋). First half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 23-Jan-2025.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦) | ||
Theorem | cofcutr2d 27652* | If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐵 is coinitial with ( R ‘𝑋). Second half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧) | ||
Theorem | cofcutrtime 27653* | If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.) |
⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)) | ||
Theorem | cofcutrtime1d 27654* | If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( L ‘𝑋) is cofinal with 𝐴. (Contributed by Scott Fenton, 23-Jan-2025.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) & ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦) | ||
Theorem | cofcutrtime2d 27655* | If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( R ‘𝑋) is coinitial with 𝐵. (Contributed by Scott Fenton, 23-Jan-2025.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) & ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧) | ||
Theorem | cofss 27656* | Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ⊆ No ) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦) | ||
Theorem | coiniss 27657* | Coinitiality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ⊆ No ) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦 ≤s 𝑥) | ||
Theorem | cutlt 27658* | Eliminating all elements below a given element of a cut does not affect the cut. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐿) ⇒ ⊢ (𝜑 → 𝐴 = (({𝑋} ∪ {𝑦 ∈ 𝐿 ∣ 𝑋 <s 𝑦}) |s 𝑅)) | ||
Theorem | cutpos 27659* | Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 0s <s 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴))) | ||
Syntax | cnorec 27660 | Declare the syntax for surreal recursion of one variable. |
class norec (𝐹) | ||
Definition | df-norec 27661* | Define the recursion generator for surreal functions of one variable. This generator creates a recursive function of surreals from their value on their left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹) | ||
Theorem | lrrecval 27662* | The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) | ||
Theorem | lrrecval2 27663* | Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) | ||
Theorem | lrrecpo 27664* | Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ 𝑅 Po No | ||
Theorem | lrrecse 27665* | Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ 𝑅 Se No | ||
Theorem | lrrecfr 27666* | Now we show that 𝑅 is founded over No . (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ 𝑅 Fr No | ||
Theorem | lrrecpred 27667* | Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} ⇒ ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) | ||
Theorem | noinds 27668* | Induction principle for a single surreal. If a property passes from a surreal's left and right sets to the surreal itself, then it holds for all surreals. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 ∈ No → (∀𝑦 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ No → 𝜒) | ||
Theorem | norecfn 27669 | Surreal recursion over one variable is a function over the surreals. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝐹 = norec (𝐺) ⇒ ⊢ 𝐹 Fn No | ||
Theorem | norecov 27670 | Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝐹 = norec (𝐺) ⇒ ⊢ (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))))) | ||
Syntax | cnorec2 27671 | Declare the syntax for surreal recursion on two arguments. |
class norec2 (𝐹) | ||
Definition | df-norec2 27672* | Define surreal recursion on two variables. This function is key to the development of most of surreal numbers. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ norec2 (𝐹) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹) | ||
Theorem | noxpordpo 27673* | To get through most of the textbook defintions in surreal numbers we will need recursion on two variables. This set of theorems sets up the preconditions for double recursion. This theorem establishes the partial ordering. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} & ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} ⇒ ⊢ 𝑆 Po ( No × No ) | ||
Theorem | noxpordfr 27674* | Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} & ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} ⇒ ⊢ 𝑆 Fr ( No × No ) | ||
Theorem | noxpordse 27675* | Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} & ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} ⇒ ⊢ 𝑆 Se ( No × No ) | ||
Theorem | noxpordpred 27676* | Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} & ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})) | ||
Theorem | no2indslem 27677* | Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.) |
⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} & ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂)) & ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑)) ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂) | ||
Theorem | no2inds 27678* | Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂)) & ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑)) ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂) | ||
Theorem | norec2fn 27679 | The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ 𝐹 = norec2 (𝐺) ⇒ ⊢ 𝐹 Fn ( No × No ) | ||
Theorem | norec2ov 27680 | The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ 𝐹 = norec2 (𝐺) ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})))) | ||
Theorem | no3inds 27681* | Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.) |
⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) & ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) & ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) & ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) & ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) & ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) & ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑)) ⇒ ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆) | ||
Syntax | cadds 27682 | Declare the syntax for surreal addition. |
class +s | ||
Definition | df-adds 27683* | Define surreal addition. This is the first of the field operations on the surreals. Definition from [Conway] p. 5. Definition from [Gonshor] p. 13. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ +s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)})))) | ||
Theorem | addsfn 27684 | Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ +s Fn ( No × No ) | ||
Theorem | addsval 27685* | The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑧 = (𝐴 +s 𝑟)}))) | ||
Theorem | addsval2 27686* | The value of surreal addition with different choices for each bound variable. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)}))) | ||
Theorem | addsrid 27687 | Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴) | ||
Theorem | addsridd 27688 | Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +s 0s ) = 𝐴) | ||
Theorem | addscom 27689 | Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴)) | ||
Theorem | addscomd 27690 | Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴)) | ||
Theorem | addslid 27691 | Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.) |
⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴) | ||
Theorem | addsproplem1 27692* | Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8403 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (( bday ‘𝐴) +no ( bday ‘𝐶))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍)))) ⇒ ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))) | ||
Theorem | addsproplem2 27693* | Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) | ||
Theorem | addsproplem3 27694* | Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))) | ||
Theorem | addsproplem4 27695* | Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍)) ⇒ ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) | ||
Theorem | addsproplem5 27696* | Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌)) ⇒ ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) | ||
Theorem | addsproplem6 27697* | Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍)) ⇒ ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) | ||
Theorem | addsproplem7 27698* | Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) ⇒ ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) | ||
Theorem | addsprop 27699 | Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.) |
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))) | ||
Theorem | addscutlem 27700* | Lemma for addscut 27701. Show the statement with some additional distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.) |
⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))) |
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