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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnpw2blenfzo2 42901 | A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → (𝑁 = (2↑((#_{b}‘𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#_{b}‘𝑁) − 1)) + 1)..^(2↑(#_{b}‘𝑁))))) | ||
Theorem | nnpw2pmod 42902 | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ → 𝑁 = ((2↑((#_{b}‘𝑁) − 1)) + (𝑁 mod (2↑((#_{b}‘𝑁) − 1))))) | ||
Theorem | blen1 42903 | The binary length of 1. (Contributed by AV, 21-May-2020.) |
⊢ (#_{b}‘1) = 1 | ||
Theorem | blen2 42904 | The binary length of 2. (Contributed by AV, 21-May-2020.) |
⊢ (#_{b}‘2) = 2 | ||
Theorem | nnpw2p 42905* | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ → ∃𝑖 ∈ ℕ_{0} ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟)) | ||
Theorem | nnpw2pb 42906* | A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ ↔ ∃𝑖 ∈ ℕ_{0} ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟)) | ||
Theorem | blen1b 42907 | The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.) |
⊢ (𝑁 ∈ ℕ_{0} → ((#_{b}‘𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1))) | ||
Theorem | blennnt2 42908 | The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.) |
⊢ (𝑁 ∈ ℕ → (#_{b}‘(2 · 𝑁)) = ((#_{b}‘𝑁) + 1)) | ||
Theorem | nnolog2flm1 42909 | The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.) |
⊢ ((𝑁 ∈ (ℤ_{≥}‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ) → (⌊‘(2 log_{b} 𝑁)) = (⌊‘(2 log_{b} (𝑁 − 1)))) | ||
Theorem | blennn0em1 42910 | The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ_{0}) → (#_{b}‘(𝑁 / 2)) = ((#_{b}‘𝑁) − 1)) | ||
Theorem | blennngt2o2 42911 | The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.) |
⊢ ((𝑁 ∈ (ℤ_{≥}‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ_{0}) → (#_{b}‘𝑁) = ((#_{b}‘((𝑁 − 1) / 2)) + 1)) | ||
Theorem | blengt1fldiv2p1 42912 | The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘2) → (#_{b}‘𝑁) = ((#_{b}‘(⌊‘(𝑁 / 2))) + 1)) | ||
Theorem | blennn0e2 42913 | The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ_{0}) → (#_{b}‘𝑁) = ((#_{b}‘(𝑁 / 2)) + 1)) | ||
Generalisation of df-bits 15348. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 42933: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)). | ||
Syntax | cdig 42914 | Extend class notation with the class of the digit extraction operation. |
class digit | ||
Definition | df-dig 42915* | Definition of an operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝑏. 𝑘 = − 1 corresponds to the first digit of the fractional part (for 𝑏 = 10 the first digit after the decimal point), 𝑘 = 0 corresponds to the last digit of the integer part (for 𝑏 = 10 the first digit before the decimal point). See also digit1 13201. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.) |
⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏))) | ||
Theorem | digfval 42916* | Operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ (𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) | ||
Theorem | digval 42917 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) | ||
Theorem | digvalnn0 42918 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ_{0}) | ||
Theorem | nn0digval 42919 | The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ_{0} ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵↑𝐾))) mod 𝐵)) | ||
Theorem | dignn0fr 42920 | The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ_{0}) ∧ 𝑁 ∈ ℕ_{0}) → (𝐾(digit‘𝐵)𝑁) = 0) | ||
Theorem | dignn0ldlem 42921 | Lemma for dignnld 42922. (Contributed by AV, 25-May-2020.) |
⊢ ((𝐵 ∈ (ℤ_{≥}‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_{≥}‘((⌊‘(𝐵 log_{b} 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾)) | ||
Theorem | dignnld 42922 | The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.) |
⊢ ((𝐵 ∈ (ℤ_{≥}‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_{≥}‘((⌊‘(𝐵 log_{b} 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0) | ||
Theorem | dig2nn0ld 42923 | The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_{≥}‘(#_{b}‘𝑁))) → (𝐾(digit‘2)𝑁) = 0) | ||
Theorem | dig2nn1st 42924 | The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.) |
⊢ (𝑁 ∈ ℕ → (((#_{b}‘𝑁) − 1)(digit‘2)𝑁) = 1) | ||
Theorem | dig0 42925 | All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0) | ||
Theorem | digexp 42926 | The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ (ℤ_{≥}‘2) ∧ 𝐾 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0}) → (𝐾(digit‘𝐵)(𝐵↑𝑁)) = if(𝐾 = 𝑁, 1, 0)) | ||
Theorem | dig1 42927 | All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ (ℤ_{≥}‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0)) | ||
Theorem | 0dig1 42928 | The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.) |
⊢ (𝐵 ∈ (ℤ_{≥}‘2) → (0(digit‘𝐵)1) = 1) | ||
Theorem | 0dig2pr01 42929 | The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.) |
⊢ (𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁) | ||
Theorem | dig2nn0 42930 | A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1}) | ||
Theorem | 0dig2nn0e 42931 | The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ (𝑁 / 2) ∈ ℕ_{0}) → (0(digit‘2)𝑁) = 0) | ||
Theorem | 0dig2nn0o 42932 | The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ ((𝑁 + 1) / 2) ∈ ℕ_{0}) → (0(digit‘2)𝑁) = 1) | ||
Theorem | dig2bits 42933 | The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐾 ∈ ℕ_{0}) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁))) | ||
Theorem | dignn0flhalflem1 42934 | Lemma 1 for dignn0flhalf 42937. (Contributed by AV, 7-Jun-2012.) |
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁)))) | ||
Theorem | dignn0flhalflem2 42935 | Lemma 2 for dignn0flhalf 42937. (Contributed by AV, 7-Jun-2012.) |
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ_{0}) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁)))) | ||
Theorem | dignn0ehalf 42936 | The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.) |
⊢ (((𝐴 / 2) ∈ ℕ_{0} ∧ 𝐴 ∈ ℕ_{0} ∧ 𝐼 ∈ ℕ_{0}) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2))) | ||
Theorem | dignn0flhalf 42937 | The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.) |
⊢ ((𝐴 ∈ (ℤ_{≥}‘2) ∧ 𝐼 ∈ ℕ_{0}) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2)))) | ||
Theorem | nn0sumshdiglemA 42938* | Lemma for nn0sumshdig 42942 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.) |
⊢ (((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ_{0} ((#_{b}‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_{b}‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglemB 42939* | Lemma for nn0sumshdig 42942 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.) |
⊢ (((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ_{0}) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ_{0} ((#_{b}‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_{b}‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglem1 42940* | Lemma 1 for nn0sumshdig 42942 (induction step). (Contributed by AV, 7-Jun-2020.) |
⊢ (𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ_{0} ((#_{b}‘𝑎) = 𝑦 → 𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ_{0} ((#_{b}‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))) | ||
Theorem | nn0sumshdiglem2 42941* | Lemma 2 for nn0sumshdig 42942. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ_{0} ((#_{b}‘𝑎) = 𝐿 → 𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘)))) | ||
Theorem | nn0sumshdig 42942* | A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝐴 ∈ ℕ_{0} → 𝐴 = Σ𝑘 ∈ (0..^(#_{b}‘𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘))) | ||
Theorem | nn0mulfsum 42943* | Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.) |
⊢ ((𝐴 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵) | ||
Theorem | nn0mullong 42944* | Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15419. (Contributed by AV, 7-Jun-2020.) |
⊢ ((𝐴 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#_{b}‘𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 42945 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | ||
Theorem | nfiund 42946 | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunord 42947* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7132, but does not use it directly, since ssorduni 7132 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunordi 42948* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ Ord 𝐵 ⇒ ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | rspcdf 42949* | Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) | ||
Theorem | spd 42950 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2417. (Contributed by Emmett Weisz, 17-Jan-2020.) |
⊢ (𝜒 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcdvw 42951* | A version of spcdv 3442 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | tfis2d 42952* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) & ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) ⇒ ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) | ||
Theorem | bnd2d 42953* | Deduction form of bnd2 8920. (Contributed by Emmett Weisz, 19-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) | ||
Theorem | dffun3f 42954* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑧𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Symbols in this section: All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 42956. The class 𝑌 is explained in the comment of setrec1lem1 42959. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 42955 | Extend class notation to include a set defined by transfinite recursion. |
class setrecs(𝐹) | ||
Definition | df-setrecs 42956* |
Define a class setrecs(𝐹) by transfinite recursion, where
(𝐹‘𝑥) is the set of new elements to add to
the class given the
set 𝑥 of elements in the class so far. We
do not need a base case,
because we can start with the empty set, which is vacuously a subset of
setrecs(𝐹). The goal of this definition is to
construct a class
fulfilling theorems setrec1 42963 and setrec2v 42968, which give a more
intuitive idea of the meaning of setrecs.
Unlike wrecs,
setrecs is well-defined for any 𝐹 and
meaningful for any
function 𝐹.
For example, see theorem onsetrec 42979 for how the class On is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤 ⊆ 𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2_{𝑜}} would fail the condition since 2_{𝑜} is not an element of the successor of ∅ or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1_{𝑜}} fails the condition, since the only subset of previously generated elements is ∅, and suc ∅ does not have 1_{𝑜} as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with ∅ and repeatedly applying the successor function. A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ∪ ran recs((𝑔 ∈ V ↦ (𝐹‘∪ ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 42963 and setrec2v 42968 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Conway noted in the Appendix to Part Zero of ONAG, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant." Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the metamath framework. The difficulty in writing a definition in metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 42963 and setrec2v 42968), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 7659, this is not actually a requirement of the metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction. If setrecs(𝐹) = ∪ 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹. One necessary condition is that each element of a given 𝑥 ∈ 𝐴 must be generated by 𝐹 when applied to a previous element 𝑦 ∈ 𝐴. In symbols, ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐴(𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐹‘𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦 ∈ 𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have ∪ 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴. The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹‘𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥∀𝑧 ∈ 𝑥¬ (𝐹‘𝑧) ⊆ 𝑦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝑦 ∈ 𝑧)} In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. ∪ {𝑦 ∣ ∀𝑧 (𝑦 ∈ 𝑧 → ∃𝑤(𝑤 ⊆ 𝑦 ∧ (𝐹‘𝑤) ∈ 𝑧 ∧ ¬ 𝑤 ∈ 𝑧))} These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦 ∈ 𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurence of ∈ 𝑧 with ⊆ 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤 ∈ 𝑧, so the definition requires the additional constraint (𝑎 ⊆ 𝑏 → (𝐹 a ) C_ ( F 𝑏)) in order to work. This issue can also be avoided by replacing ∈ 𝑧 with ⊆ 𝑧. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(𝑦 ∩ 𝑧 ≠ ∅ → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∩ 𝑧 ≠ ∅))} In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹‘𝑤) for 𝑤 ⊆ (𝑦 ∩ 𝑧) even if those elements of (𝐹‘𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹‘𝑤) in the definition by ((𝐹‘𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦 ⊆ 𝑧 would be false. We could then simply add all elements of (𝐹‘𝑤) outside of 𝑦 for any 𝑤 ⊆ 𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the metamath definition. The other direction is easier. If a certain 𝑧 fails the metamath definition, then all (𝐹‘𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦 ∩ 𝑧), and in particular ((𝐹‘𝑤) ∩ 𝑦) ⊆ 𝑧. The foundedness version is starting to look more like ax-reg 8653! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣 ∈ 𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ 𝑣 ∈ (𝐹‘𝑤)))} Now, if we replace (𝐹‘𝑤) by ((𝐹‘𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣 ∈ 𝑦 since 𝑣 ∈ 𝑧 and 𝑧 ⊆ 𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5208, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧. Version 4 - Foundedness (6-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤∀𝑢 ∈ 𝑧(𝑤 ⊆ 𝑦 ∧ ¬ 𝑢 ∈ 𝑤 ∧ 𝑣 ∈ (𝐹‘𝑤)) This is so close to df-fr 5208; the only change needed is to switch ∃𝑤 with ∀𝑢 ∈ 𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 8653. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of ∪ 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦 ⊆ 𝑧 rather than only 𝑦 ∈ 𝑧. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(∃𝑎(𝑎 ⊆ 𝑦 ∧ 𝑎 ∈ 𝑧) → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. ) Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of ∪ 𝑧. I think this has the same problem. We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} Now this looks more like df-fr 5208! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹‘𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5208. This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧. Version 1.3 (31-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝑤 ∈ 𝑧 ∧ (𝐹‘𝑤) ∈ 𝑧))) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝑤 ∈ 𝑧 ∨ (𝐹‘𝑤) ∈ 𝑧)))} 𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work. Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹‘𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹‘𝑤) corresponds to the variable 𝑦 in df-fr 5208. Thus, in our case, df-fr 5208 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧(𝐹‘𝑤) ∈ (𝐹‘𝑣))) This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑣 ∈ 𝑧¬ 𝑣𝑅(𝐹‘𝑤) so we can replace one with the other. Negate both sides: 𝑤 ∈ 𝑧 ↔ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤) If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.) |
⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | ||
Theorem | setrecseq 42957 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺)) | ||
Theorem | nfsetrecs 42958 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥setrecs(𝐹) | ||
Theorem | setrec1lem1 42959* |
Lemma for setrec1 42963. This is a utility theorem showing the
equivalence
of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof
uses
elabg 3502 and equivalence theorems.
Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) | ||
Theorem | setrec1lem2 42960* | Lemma for setrec1 42963. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3741, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌) | ||
Theorem | setrec1lem3 42961* | Lemma for setrec1 42963. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 42960. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴 ∈ 𝑌. I don't know if proving this fact directly using setrec1lem1 42959 would be any easier than the current proof using setrec1lem2 42960, and it would only slightly simplify the proof of setrec1 42963. Other than the use of bnd2d 42953, this is a purely technical theorem for rearranging notation from that of setrec1lem2 42960 to that of setrec1 42963. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌)) ⇒ ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌)) | ||
Theorem | setrec1lem4 42962* |
Lemma for setrec1 42963. If 𝑋 is recursively generated by 𝐹, then
so is 𝑋 ∪ (𝐹‘𝐴).
In the proof of setrec1 42963, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis Ⅎ𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1910, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 42960 could similarly be made easier to read by adding the hypothesis Ⅎ𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
⊢ Ⅎ𝑧𝜑 & ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌) | ||
Theorem | setrec1 42963 |
This is the first of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is closed under 𝐹. This
effectively sets the
actual value of setrecs(𝐹) as a lower bound for
setrecs(𝐹), as it implies that any set
generated by successive
applications of 𝐹 is a member of 𝐵. This
theorem "gets off the
ground" because we can start by letting 𝐴 = ∅, and the
hypotheses
of the theorem will hold trivially.
Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon). Proof summary: Assume that 𝐴 ⊆ 𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 42961, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 42961.) Therefore, by setrec1lem4 42962, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4596, it is a subset of the union of all sets recursively generated by 𝐹. See df-setrecs 42956 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵) | ||
Theorem | setrec2fun 42964* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, theorems setrec1 42963 and setrec2v 42968 say
that setrecs(𝐹) is the minimal class closed under
𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7200) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ Fun 𝐹 & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2lem1 42965* | Lemma for setrec2 42967. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) | ||
Theorem | setrec2lem2 42966* | Lemma for setrec2 42967. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) | ||
Theorem | setrec2 42967* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, theorems setrec1 42963 and setrec2v 42968
uniquely determine setrecs(𝐹) to be the minimal class closed
under 𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7200) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2v 42968* | Version of setrec2 42967 with a dv condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setis 42969* | Version of setrec2 42967 expressed as an induction schema. This theorem is a generalization of tfis3 7204. (Contributed by Emmett Weisz, 27-Feb-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) | ||
Theorem | elsetrecslem 42970* | Lemma for elsetrecs 42971. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 42968. To see why this lemma also requires setrec1 42963, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | elsetrecs 42971* | A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 42963 and setrec2 42967, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | setrecsss 42972 | The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.) |
⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐹 ⊆ 𝐺) ⇒ ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) | ||
Theorem | setrecsres 42973 | A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) | ||
Theorem | vsetrec 42974 | Construct V using set recursion. The proof indirectly uses trcl 8768, which relies on rec, but theoretically 𝐶 in trcl 8768 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) ⇒ ⊢ setrecs(𝐹) = V | ||
Theorem | 0setrec 42975 | If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ (𝜑 → (𝐹‘∅) = ∅) ⇒ ⊢ (𝜑 → setrecs(𝐹) = ∅) | ||
Theorem | onsetreclem1 42976* | Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} | ||
Theorem | onsetreclem2 42977* | Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) | ||
Theorem | onsetreclem3 42978* | Lemma for onsetrec 42979. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) | ||
Theorem | onsetrec 42979 |
Construct On using set recursion. When 𝑥 ∈
On, the function
𝐹 constructs the least ordinal greater
than any of the elements of
𝑥, which is ∪ 𝑥 for a limit ordinal and suc ∪ 𝑥 for a
successor ordinal.
For example, (𝐹‘{1_{𝑜}, 2_{𝑜}}) = {∪ {1_{𝑜}, 2_{𝑜}}, suc ∪ {1_{𝑜}, 2_{𝑜}}} = {2_{𝑜}, 3_{𝑜}} which contains 3_{𝑜}, and (𝐹‘ω) = {∪ ω, suc ∪ ω} = {ω, ω +_{𝑜} 1_{𝑜}}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated. Any function 𝐹 fulfilling lemmas onsetreclem2 42977 and onsetreclem3 42978 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc ∪ 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 32029. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ setrecs(𝐹) = On | ||
Model organization after organization of reals - see TOC | ||
Syntax | cpg 42980 | Extend class notation to include the class of partizan game forms. |
class Pg | ||
Definition | df-pg 42981 | Define the class of partizan games. More precisely, this is the class of partizan game forms, many of which represent equal partisan games. In metamath, equality between partizan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.) |
⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥))) | ||
Theorem | elpglem1 42982* | Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥)) → ((1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg)) | ||
Theorem | elpglem2 42983* | Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (((1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥))) | ||
Theorem | elpglem3 42984* | Lemma for elpg 42985. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1^{st} ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^{nd} ‘𝐴) ∈ 𝒫 𝑥)))) | ||
Theorem | elpg 42985 | Membership in the class of partizan games. In ONAG this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿 ∣ 𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1^{st} ‘𝐴) ⊆ Pg ∧ (2^{nd} ‘𝐴) ⊆ Pg)) | ||
This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/. | ||
Theorem | 19.8ad 42986 | If a wff is true, it is true for at least one instance. Deductive form of 19.8a 2206. (Contributed by DAW, 13-Feb-2017.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | sbidd 42987 | An identity theorem for substitution. See sbid 2270. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | sbidd-misc 42988 | An identity theorem for substitution. See sbid 2270. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓)) | ||
As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems. | ||
Syntax | cge-real 42989 | Extend wff notation to include the 'greater than or equal to' relation, see df-gte 42991. |
class ≥ | ||
Syntax | cgt 42990 | Extend wff notation to include the 'greater than' relation, see df-gt 42992. |
class > | ||
Definition | df-gte 42991 |
Define the 'greater than or equal' predicate over the reals. Defined in
ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the
"NIST Digital Library of Mathematical Functions" , front
introduction,
"Common Notations and Definitions" section at
http://dlmf.nist.gov/front/introduction#Sx4.
This relation is merely
the converse of the 'less than or equal to' relation defined by df-le 10282.
We do not write this as (𝑥 ≥ 𝑦 ↔ 𝑦 ≤ 𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦 ↔ 𝑦 < 𝑥), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: ⊢ > = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ^{*} ∧ 𝑦 ∈ ℝ^{*}) ∧ 𝑦 < 𝑥)} and ⊢ ≥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ^{*} ∧ 𝑦 ∈ ℝ^{*}) ∧ 𝑦 ≤ 𝑥)} but these are very complicated. This definition of ≥, and the similar one for > (df-gt 42992), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 42993 for a more conventional expression of the relationship between < and >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ≥ = ^{◡} ≤ | ||
Definition | df-gt 42992 |
The 'greater than' relation is merely the converse of the 'less than or
equal to' relation defined by df-lt 10151. Defined in ISO 80000-2:2009(E)
operation 2-7.12. See df-gte 42991 for a discussion on why this approach is
used for the definition. See gt-lt 42994 and gt-lth 42996 for more conventional
expression of the relationship between < and
>.
As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ > = ^{◡} < | ||
Theorem | gte-lte 42993 | Simple relationship between ≤ and ≥. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | gt-lt 42994 | Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)) | ||
Theorem | gte-lteh 42995 | Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) | ||
Theorem | gt-lth 42996 | Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴) | ||
Theorem | ex-gt 42997 | Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ ¬ 0 > 0 | ||
Theorem | ex-gte 42998 | Simple example of ≥, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ 0 ≥ 0 | ||
It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved. | ||
Syntax | csinh 42999 | Extend class notation to include the hyperbolic sine function, see df-sinh 43002. |
class sinh | ||
Syntax | ccosh 43000 | Extend class notation to include the hyperbolic cosine function. see df-cosh 43003. |
class cosh |
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