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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nosgnn0i 32401 | If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.) |
⊢ 𝑋 ∈ {1o, 2o} ⇒ ⊢ ∅ ≠ 𝑋 | ||
Theorem | noreson 32402 | The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No ) | ||
Theorem | sltintdifex 32403* | If 𝐴 <s 𝐵, then the intersection of all the ordinals that have differing signs in 𝐴 and 𝐵 exists. (Contributed by Scott Fenton, 22-Feb-2012.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ V)) | ||
Theorem | sltres 32404 | If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → 𝐴 <s 𝐵)) | ||
Theorem | noxp1o 32405 | The Cartesian product of an ordinal and {1o} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.) |
⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) | ||
Theorem | noseponlem 32406* | Lemma for nosepon 32407. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥)) | ||
Theorem | nosepon 32407* | Given two unequal surreals, the minimal ordinal at which they differ is an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On) | ||
Theorem | noextend 32408 | Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
⊢ 𝑋 ∈ {1o, 2o} ⇒ ⊢ (𝐴 ∈ No → (𝐴 ∪ {〈dom 𝐴, 𝑋〉}) ∈ No ) | ||
Theorem | noextendseq 32409 | Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.) |
⊢ 𝑋 ∈ {1o, 2o} ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No ) | ||
Theorem | noextenddif 32410* | Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.) |
⊢ 𝑋 ∈ {1o, 2o} ⇒ ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {〈dom 𝐴, 𝑋〉})‘𝑥)} = dom 𝐴) | ||
Theorem | noextendlt 32411 | Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
⊢ (𝐴 ∈ No → (𝐴 ∪ {〈dom 𝐴, 1o〉}) <s 𝐴) | ||
Theorem | noextendgt 32412 | Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {〈dom 𝐴, 2o〉})) | ||
Theorem | nolesgn2o 32413 | Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2o) | ||
Theorem | nolesgn2ores 32414 | Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋)) | ||
Theorem | sltsolem1 32415 | Lemma for sltso 32416. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ {〈1o, ∅〉, 〈1o, 2o〉, 〈∅, 2o〉} Or ({1o, 2o} ∪ {∅}) | ||
Theorem | sltso 32416 | Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.) |
⊢ <s Or No | ||
Theorem | bdayfo 32417 | The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.) |
⊢ bday : No –onto→On | ||
Theorem | fvnobday 32418 | The value of a surreal at its birthday is ∅. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) |
⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅) | ||
Theorem | nosepnelem 32419* | Lemma for nosepne 32420. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | ||
Theorem | nosepne 32420* | The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | ||
Theorem | nosep1o 32421* | If the value of a surreal at a separator is 1o then the surreal is lesser. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → 𝐴 <s 𝐵) | ||
Theorem | nosepdmlem 32422* | Lemma for nosepdm 32423. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)) | ||
Theorem | nosepdm 32423* | The first place two surreals differ is an element of the larger of their domains. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)) | ||
Theorem | nosepeq 32424* | The values of two surreals at a point less than their separators are equal. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ 𝑋 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘𝑋) = (𝐵‘𝑋)) | ||
Theorem | nosepssdm 32425* | Given two non-equal surreals, their separator is less than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴) | ||
Theorem | nodenselem4 32426* | Lemma for nodense 32431. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On) | ||
Theorem | nodenselem5 32427* | Lemma for nodense 32431. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 32426 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)) | ||
Theorem | nodenselem6 32428* | The restriction of a surreal to the abstraction from nodenselem4 32426 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No ) | ||
Theorem | nodenselem7 32429* | Lemma for nodense 32431. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐴‘𝐶) = (𝐵‘𝐶))) | ||
Theorem | nodenselem8 32430* | Lemma for nodense 32431. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o))) | ||
Theorem | nodense 32431* | Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵)) | ||
Theorem | bdayimaon 32432 | Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.) |
⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On) | ||
Theorem | nolt02olem 32433 | Lemma for nolt02o 32434. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋) | ||
Theorem | nolt02o 32434 | Given 𝐴 less than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = 2o) | ||
Theorem | noresle 32435* | Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈) | ||
Theorem | nomaxmo 32436* | A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) | ||
Theorem | noprefixmo 32437* | In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.) |
⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥)) | ||
Theorem | nosupno 32438* | The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No ) | ||
Theorem | nosupdm 32439* | The domain of the surreal supremum when there is no maximum. The primary point of this theorem is to change bound variable. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}) | ||
Theorem | nosupbday 32440* | Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) ⊆ suc ∪ ( bday “ 𝐴)) | ||
Theorem | nosupfv 32441* | The value of surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆‘𝐺) = (𝑈‘𝐺)) | ||
Theorem | nosupres 32442* | A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺)) | ||
Theorem | nosupbnd1lem1 32443* | Lemma for nosupbnd1 32449. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆)) | ||
Theorem | nosupbnd1lem2 32444* | Lemma for nosupbnd1 32449. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆) | ||
Theorem | nosupbnd1lem3 32445* | Lemma for nosupbnd1 32449. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 2o. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o) | ||
Theorem | nosupbnd1lem4 32446* | Lemma for nosupbnd1 32449. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not undefined. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅) | ||
Theorem | nosupbnd1lem5 32447* | Lemma for nosupbnd1 32449. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 1o. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o) | ||
Theorem | nosupbnd1lem6 32448* | Lemma for nosupbnd1 32449. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆) | ||
Theorem | nosupbnd1 32449* | Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆) | ||
Theorem | nosupbnd2lem1 32450* | Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {〈dom 𝑈, 2o〉})) | ||
Theorem | nosupbnd2 32451* | Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)) | ||
Theorem | noetalem1 32452* | Lemma for noeta 32457. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No ) | ||
Theorem | noetalem2 32453* | Lemma for noeta 32457. 𝑍 is an upper bound for 𝐴. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 4-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑋 ∈ 𝐴) → 𝑋 <s 𝑍) | ||
Theorem | noetalem3 32454* | Lemma for noeta 32457. When 𝐴 and 𝐵 are separated, then 𝑍 is a lower bound for 𝐵. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) | ||
Theorem | noetalem4 32455* | Lemma for noeta 32457. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) | ||
Theorem | noetalem5 32456* | Lemma for noeta 32457. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
Theorem | noeta 32457* | The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, there is an upper bound on the birthday of that surreal. Alling's axiom FE. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
Syntax | csle 32458 | Declare the syntax for surreal less than or equal. |
class ≤s | ||
Definition | df-sle 32459 | Define the surreal less than or equal predicate. Compare df-le 10417. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ≤s = (( No × No ) ∖ ◡ <s ) | ||
Theorem | sltirr 32460 | Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴) | ||
Theorem | slttr 32461 | Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | ||
Theorem | sltasym 32462 | Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴)) | ||
Theorem | sltlin 32463 | Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴)) | ||
Theorem | slttrieq2 32464 | Trichotomy law for surreal less than. (Contributed by Scott Fenton, 22-Apr-2012.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴))) | ||
Theorem | slttrine 32465 | Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴))) | ||
Theorem | slenlt 32466 | Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | ||
Theorem | sltnle 32467 | Surreal less than in terms of less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴)) | ||
Theorem | sleloe 32468 | Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | sletri3 32469 | Trichotomy law for surreal less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴))) | ||
Theorem | sltletr 32470 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 <s 𝐶)) | ||
Theorem | slelttr 32471 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | ||
Theorem | sletr 32472 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) | ||
Theorem | slttrd 32473 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → 𝐵 <s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
Theorem | sltletrd 32474 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → 𝐵 ≤s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
Theorem | slelttrd 32475 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 ≤s 𝐵) & ⊢ (𝜑 → 𝐵 <s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
Theorem | sletrd 32476 | Surreal less than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 ≤s 𝐵) & ⊢ (𝜑 → 𝐵 ≤s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤s 𝐶) | ||
Theorem | bdayfun 32477 | The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ Fun bday | ||
Theorem | bdayfn 32478 | The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.) |
⊢ bday Fn No | ||
Theorem | bdaydm 32479 | The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ dom bday = No | ||
Theorem | bdayrn 32480 | The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ ran bday = On | ||
Theorem | bdayelon 32481 | The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.) |
⊢ ( bday ‘𝐴) ∈ On | ||
Theorem | nocvxminlem 32482* | Lemma for nocvxmin 32483. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.) |
⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ¬ 𝑋 <s 𝑌)) | ||
Theorem | nocvxmin 32483* | Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)) | ||
Theorem | noprc 32484 | The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ¬ No ∈ V | ||
Syntax | csslt 32485 | Declare the syntax for surreal set less than. |
class <<s | ||
Definition | df-sslt 32486* | Define the relationship that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ <<s = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)} | ||
Syntax | cscut 32487 | Declare the syntax for the surreal cut operator. |
class |s | ||
Definition | df-scut 32488* | Define the cut operator on surreal numbers. This operator, which Conway takes as the primitive operator over surreals, picks the surreal lying between two sets of surreals of minimal birthday. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))) | ||
Theorem | brsslt 32489* | Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | ||
Theorem | ssltex1 32490 | The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssltex2 32491 | The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | ||
Theorem | ssltss1 32492 | The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | ||
Theorem | ssltss2 32493 | The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | ||
Theorem | ssltsep 32494* | The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | ||
Theorem | sssslt1 32495 | Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵) | ||
Theorem | sssslt2 32496 | Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) | ||
Theorem | nulsslt 32497 | The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) | ||
Theorem | nulssgt 32498 | The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) | ||
Theorem | conway 32499* | Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) | ||
Theorem | scutval 32500* | The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))) |
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