HomeTheorem List (p. 280 of 503)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | madeun 27901 | 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 27902 | 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 27903 | 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 27904 | 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 27905 | 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 27906 | 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 27907 | No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.) |
| ⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋)) | ||
| Theorem | leftirr 27908 | No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.) |
| ⊢ ¬ 𝑋 ∈ ( L ‘𝑋) | ||
| Theorem | rightirr 27909 | No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.) |
| ⊢ ¬ 𝑋 ∈ ( R ‘𝑋) | ||
| Theorem | left0s 27910 | The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.) |
| ⊢ ( L ‘ 0s ) = ∅ | ||
| Theorem | right0s 27911 | The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.) |
| ⊢ ( R ‘ 0s ) = ∅ | ||
| Theorem | left1s 27912 | The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.) |
| ⊢ ( L ‘ 1s ) = { 0s } | ||
| Theorem | right1s 27913 | The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.) |
| ⊢ ( R ‘ 1s ) = ∅ | ||
| Theorem | lrold 27914 | 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 27915* | Lemma for madebday 27917. If the inductive hypothesis of madebday 27917 is satisfied, the converse of oldbdayim 27906 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 27916* | Lemma for madebday 27917. If the inductive hypothesis of madebday 27917 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27921 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 27917 | 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 27918 | 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 27919 | 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 | newbdayim 27920 | One direction of the biconditional in newbday 27919. (Contributed by Scott Fenton, 7-Nov-2025.) |
| ⊢ (𝑋 ∈ ( N ‘𝐴) → ( bday ‘𝑋) = 𝐴) | ||
| Theorem | lrcut 27921 | 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 | cutsfo 27922 | The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.) |
| ⊢ |s : <<s –onto→ No | ||
| Theorem | ltsn0 27923 | 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 27924 | 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 | ltslpss 27925 | 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 | leslss 27926 | 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 27927 | Zero is in the old set of any nonzero number. (Contributed by Scott Fenton, 13-Mar-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐴 ≠ 0s ) ⇒ ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴))) | ||
| Theorem | 0elleft 27928 | 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 27929 | 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 | madefi 27930 | The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.) |
| ⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin) | ||
| Theorem | oldfi 27931 | The old set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.) |
| ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin) | ||
| Theorem | bdayiun 27932* | The birthday of a surreal is the least upper bound of the successors of the birthdays of its options. This is the definition of the birthday of a combinatorial game in the Lean Combinatorial Game Theory library at https://github.com/vihdzp/combinatorial-games. (Contributed by Scott Fenton, 22-Nov-2025.) |
| ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥)) | ||
| Theorem | bdayle 27933* | A condition for bounding a birthday above. (Contributed by Scott Fenton, 22-Nov-2025.) |
| ⊢ ((𝑋 ∈ No ∧ Ord 𝑂) → (( bday ‘𝑋) ⊆ 𝑂 ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑋))( bday ‘𝑦) ∈ 𝑂)) | ||
| Theorem | sltsbday 27934 | Birthday comparison rule for surreals. (Contributed by Scott Fenton, 23-Feb-2026.) |
| ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐿 <<s {𝐵}) & ⊢ (𝜑 → {𝐵} <<s 𝑅) ⇒ ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵)) | ||
| Theorem | cofslts 27935* | 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 | coinitslts 27936* | 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 27937* | 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 27938* | 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 27939* | 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 27940* | 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 27941* | 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 27942* | 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 27943* | 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 27944* | 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 27945* | 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 27946* | 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 27947* | 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 27948* | 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 27949* | 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 27950* | 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))) | ||
| Theorem | cutmax 27951* | If 𝐴 has a maximum, then the maximum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.) |
| ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋) ⇒ ⊢ (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵)) | ||
| Theorem | cutmin 27952* | If 𝐵 has a minimum, then the minimum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.) |
| ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝑋 ≤s 𝑦) ⇒ ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐴 |s {𝑋})) | ||
| Theorem | cutminmax 27953* | If the left set of 𝑋 has a maximum and the right set of 𝑋 has a minimum, then 𝑋 is equal to the cut of the maximum and the minimum. (Contributed by Scott Fenton, 23-Feb-2026.) |
| ⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋)) & ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿) & ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝑋)) & ⊢ (𝜑 → ∀𝑦 ∈ ( R ‘𝑋)𝑅 ≤s 𝑦) ⇒ ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅})) | ||
| Syntax | cnorec 27954 | Declare the syntax for surreal recursion of one variable. |
| class norec (𝐹) | ||
| Definition | df-norec 27955* | 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 27956* | 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 27957* | 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 27958* | 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 27959* | 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 27960* | 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 27961* | 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 27962* | 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 27963 | 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 27964 | 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 27965 | Declare the syntax for surreal recursion on two arguments. |
| class norec2 (𝐹) | ||
| Definition | df-norec2 27966* | 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 27967* | To get through most of the textbook definitions 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 27968* | 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 27969* | 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 27970* | 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 | no2indlesm 27971* | 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 27972* | 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 27973 | 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 27974 | 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 27975* | 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 27976 | Declare the syntax for surreal addition. |
| class +s | ||
| Definition | df-adds 27977* | 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 27978 | Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.) |
| ⊢ +s Fn ( No × No ) | ||
| Theorem | addsval 27979* | 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 27980* | 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 27981 | 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 27982 | 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 27983 | 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 27984 | 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 27985 | Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.) |
| ⊢ (𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴) | ||
| Theorem | addsproplem1 27986* | 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 8335 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 27987* | 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 27988* | 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 27989* | 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 27990* | 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 27991* | 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 27992* | 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 27993 | 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 | addcutslem 27994* | Lemma for addcuts 27995. 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)}))) | ||
| Theorem | addcuts 27995* | Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-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)}))) | ||
| Theorem | addcuts2 27996* | Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025.) |
| ⊢ (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 | addscld 27997 | Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
| ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No ) | ||
| Theorem | addscl 27998 | Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No ) | ||
| Theorem | addsf 27999 | Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
| ⊢ +s :( No × No )⟶ No | ||
| Theorem | addsfo 28000 | Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.) |
| ⊢ +s :( No × No )–onto→ No | ||
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