P. 172 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	decsplit1 17101	Split a decimal number into two parts. Base case: 𝑁 = 1. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((𝐴 · (;10↑1)) + 𝐵) = ;𝐴𝐵
Theorem	decsplit 17102	Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ (𝑀 + 1) = 𝑁    &   ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶    ⇒   ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷
Theorem	karatsuba 17103	The Karatsuba multiplication algorithm. If 𝑋 and 𝑌 are decomposed into two groups of digits of length 𝑀 (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then 𝑋𝑌 can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 12773. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝑆 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ (𝐴 · 𝐶) = 𝑅    &   ⊢ (𝐵 · 𝐷) = 𝑇    &   ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = ((𝑅 + 𝑆) + 𝑇)    &   ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝑋    &   ⊢ ((𝐶 · (;10↑𝑀)) + 𝐷) = 𝑌    &   ⊢ ((𝑅 · (;10↑𝑀)) + 𝑆) = 𝑊    &   ⊢ ((𝑊 · (;10↑𝑀)) + 𝑇) = 𝑍    ⇒   ⊢ (𝑋 · 𝑌) = 𝑍
Theorem	2exp4 17104	Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
⊢ (2↑4) = ;16
Theorem	2exp5 17105	Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
⊢ (2↑5) = ;32
Theorem	2exp6 17106	Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (2↑6) = ;64
Theorem	2exp7 17107	Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
⊢ (2↑7) = ;;128
Theorem	2exp8 17108	Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
⊢ (2↑8) = ;;256
Theorem	2exp11 17109	Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
⊢ (2↑;11) = ;;;2048
Theorem	2exp16 17110	Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
⊢ (2↑;16) = ;;;;65536
Theorem	3exp3 17111	Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
⊢ (3↑3) = ;27
Theorem	2expltfac 17112	The factorial grows faster than two to the power 𝑁. (Contributed by Mario Carneiro, 15-Sep-2016.)
⊢ (𝑁 ∈ (ℤ≥‘4) → (2↑𝑁) < (!‘𝑁))
6.2.18  Cyclical shifts of words (cont.)
Theorem	cshwsidrepsw 17113	If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
Theorem	cshwsidrepswmod0 17114	If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)))))
Theorem	cshwshashlem1 17115*	If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)
⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))    ⇒   ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(♯‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊)
Theorem	cshwshashlem2 17116*	If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))    ⇒   ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
Theorem	cshwshashlem3 17117*	If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))    ⇒   ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ≠ 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
Theorem	cshwsdisj 17118*	The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))    ⇒   ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
Theorem	cshwsiun 17119*	The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
Theorem	cshwsex 17120*	The class of (different!) words resulting by cyclically shifting a given word is a set. (Contributed by AV, 8-Jun-2018.) (Revised by AV, 8-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 ∈ V)
Theorem	cshws0 17121*	The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ (𝑊 = ∅ → (♯‘𝑀) = 0)
Theorem	cshwrepswhash1 17122*	The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)
Theorem	cshwshashnsame 17123*	If a word (not consisting of identical symbols) has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (♯‘𝑀) = (♯‘𝑊)))
Theorem	cshwshash 17124*	If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((♯‘𝑀) = (♯‘𝑊) ∨ (♯‘𝑀) = 1))
6.2.19  Specific prime numbers
Theorem	prmlem0 17125*	Lemma for prmlem1 17127 and prmlem2 17139. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ ((¬ 2 ∥ 𝑀 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))    &   ⊢ (𝐾 ∈ ℙ → ¬ 𝐾 ∥ 𝑁)    &   ⊢ (𝐾 + 2) = 𝑀    ⇒   ⊢ ((¬ 2 ∥ 𝐾 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
Theorem	prmlem1a 17126*	A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 1 < 𝑁    &   ⊢ ¬ 2 ∥ 𝑁    &   ⊢ ¬ 3 ∥ 𝑁    &   ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	prmlem1 17127	A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 1 < 𝑁    &   ⊢ ¬ 2 ∥ 𝑁    &   ⊢ ¬ 3 ∥ 𝑁    &   ⊢ 𝑁 < ;25    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	5prm 17128	5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ 5 ∈ ℙ
Theorem	6nprm 17129	6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ ¬ 6 ∈ ℙ
Theorem	7prm 17130	7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ 7 ∈ ℙ
Theorem	8nprm 17131	8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ ¬ 8 ∈ ℙ
Theorem	9nprm 17132	9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ ¬ 9 ∈ ℙ
Theorem	10nprm 17133	10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ ¬ ;10 ∈ ℙ
Theorem	11prm 17134	11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ ;11 ∈ ℙ
Theorem	13prm 17135	13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ ;13 ∈ ℙ
Theorem	17prm 17136	17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ ;17 ∈ ℙ
Theorem	19prm 17137	19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ ;19 ∈ ℙ
Theorem	23prm 17138	23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ ;23 ∈ ℙ
Theorem	prmlem2 17139	Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows to cover the numbers less than 5↑2 = 25. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to 29↑2 = 841, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 17155). As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 < ;;841    &   ⊢ 1 < 𝑁    &   ⊢ ¬ 2 ∥ 𝑁    &   ⊢ ¬ 3 ∥ 𝑁    &   ⊢ ¬ 5 ∥ 𝑁    &   ⊢ ¬ 7 ∥ 𝑁    &   ⊢ ¬ ;11 ∥ 𝑁    &   ⊢ ¬ ;13 ∥ 𝑁    &   ⊢ ¬ ;17 ∥ 𝑁    &   ⊢ ¬ ;19 ∥ 𝑁    &   ⊢ ¬ ;23 ∥ 𝑁    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	37prm 17140	37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;37 ∈ ℙ
Theorem	43prm 17141	43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;43 ∈ ℙ
Theorem	83prm 17142	83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;83 ∈ ℙ
Theorem	139prm 17143	139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;139 ∈ ℙ
Theorem	163prm 17144	163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;163 ∈ ℙ
Theorem	317prm 17145	317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;317 ∈ ℙ
Theorem	631prm 17146	631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;631 ∈ ℙ
Theorem	prmo4 17147	The primorial of 4. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘4) = 6
Theorem	prmo5 17148	The primorial of 5. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘5) = ;30
Theorem	prmo6 17149	The primorial of 6. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘6) = ;30
6.2.20  Very large primes
Theorem	1259lem1 17150	Lemma for 1259prm 17155. Calculate a power mod. In decimal, we calculate 2↑16 = 52𝑁 + 68≡68 and 2↑17≡68 · 2 = 136 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
Theorem	1259lem2 17151	Lemma for 1259prm 17155. Calculate a power mod. In decimal, we calculate 2↑34 = (2↑17)↑2≡136↑2≡14𝑁 + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
Theorem	1259lem3 17152	Lemma for 1259prm 17155. Calculate a power mod. In decimal, we calculate 2↑38 = 2↑34 · 2↑4≡870 · 16 = 11𝑁 + 71 and 2↑76 = (2↑34)↑2≡71↑2 = 4𝑁 + 5≡5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
Theorem	1259lem4 17153	Lemma for 1259prm 17155. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	1259lem5 17154	Lemma for 1259prm 17155. Calculate the GCD of 2↑34 − 1≡869 with 𝑁 = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
Theorem	1259prm 17155	1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	2503lem1 17156	Lemma for 2503prm 17159. Calculate a power mod. In decimal, we calculate 2↑18 = 512↑2 = 104𝑁 + 1832≡1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
Theorem	2503lem2 17157	Lemma for 2503prm 17159. Calculate a power mod. We calculate 2↑19 = 2↑18 · 2≡1832 · 2 = 𝑁 + 1161, 2↑38 = (2↑19)↑2≡1161↑2 = 538𝑁 + 1307, 2↑39 = 2↑38 · 2≡1307 · 2 = 𝑁 + 111, 2↑78 = (2↑39)↑2≡111↑2 = 5𝑁 − 194, 2↑156 = (2↑78)↑2≡194↑2 = 15𝑁 + 91, 2↑312 = (2↑156)↑2≡91↑2 = 3𝑁 + 772, 2↑624 = (2↑312)↑2≡772↑2 = 238𝑁 + 270, 2↑1248 = (2↑624)↑2≡270↑2 = 29𝑁 + 313, 2↑1251 = 2↑1248 · 8≡313 · 8 = 𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1251)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	2503lem3 17158	Lemma for 2503prm 17159. Calculate the GCD of 2↑18 − 1≡1831 with 𝑁 = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
Theorem	2503prm 17159	2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	4001lem1 17160	Lemma for 4001prm 17164. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
Theorem	4001lem2 17161	Lemma for 4001prm 17164. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
Theorem	4001lem3 17162	Lemma for 4001prm 17164. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑4≡1↑4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	4001lem4 17163	Lemma for 4001prm 17164. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1
Theorem	4001prm 17164	4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ 𝑁 ∈ ℙ
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.1.1  Basic definitions An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit. An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of ℕ. The function's argument is the index of a structure component (such as 1 for the base set of a group), and its value is the component (such as the base set). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 17216 and strfv 17222. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 17216, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using the extensible structure {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨;10, 𝐿⟩}. See section header comment mmtheorems.html#cnx 17216 for more details on numeric indices versus the structure component extractors. There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an 𝑋 is a 𝑌 via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach. To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-ress 17252. This can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone. Individual kinds of structures will need to handle this behavior by ignoring operators' values outside the range (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp). For example, the unital ring of integers ℤring is defined in df-zring 21408 as simply ℤring = (ℂfld ↾s ℤ). This can be similarly done for all other subsets of ℂ, which has all the structure we can show applies to it, and this all comes "for free". Should we come up with some new structure in the future that we wish ℂ to inherit, then we change the definition of ℂfld, reprove all the slot extraction theorems, add a new one, and that's it. None of the other downstream theorems have to change. Note that the construct of df-prds 17461 addresses a different situation. It is not possible to have SubGrp and SubRing be the same thing because they produce different outputs on the same input. The subgroups of an extensible structure treated as a group are not the same as the subrings of that same structure. With df-prds 17461 it can actually reasonably perform the task, that is, being the product group given a family of groups, while also being the product ring given a family of rings. There is no contradiction here because the group part of a product ring is a product group. There is also a general theory of "substructure algebras", in the form of df-mre 17598 and df-acs 17601. SubGrp is a Moore collection, as is SubRing, SubRng and many other substructure collections. But it is not useful for picking out a particular collection of interest; SubRing and SubGrp still need to be defined and they are distinct --- nothing is going to select these definitions for us. Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.
7.1.1.1  Extensible structures as structures with components
Syntax	cstr 17165	Extend class notation with the class of structures with components numbered below 𝐴.
class Struct
Definition	df-struct 17166*	Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set ∅ to be extensible structures. Because of 0nelfun 6554, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17170: 𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}). Allowing an extensible structure to contain the empty set ensures that expressions like {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} are structures without asserting or implying that 𝐴, 𝐵, 𝐶 and 𝐷 are sets (if 𝐴 or 𝐵 is a proper class, then ⟨𝐴, 𝐵⟩ = ∅, see opprc 4872). This is used critically in strle1 17177, strle2 17178, strle3 17179 and strleun 17176 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17350 and theorems like it will have many sethood assumptions, and may not even be usable in the empty context. Instead, the sethood assumption is deferred until it is actually needed, e.g., ipsbase 17351, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like ℂfld does not contain the empty set, and therefore is a function, see cnfldfun 21329. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
Theorem	brstruct 17167	The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ Rel Struct
Theorem	isstruct2 17168	The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
Theorem	structex 17169	A structure is a set. (Contributed by AV, 10-Nov-2021.)
⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
Theorem	structn0fun 17170	A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))
Theorem	isstruct 17171	The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
Theorem	structcnvcnv 17172	Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅}))
Theorem	structfung 17173	The converse of the converse of a structure is a function. Closed form of structfun 17174. (Contributed by AV, 12-Nov-2021.)
⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹)
Theorem	structfun 17174	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
⊢ 𝐹 Struct 𝑋    ⇒   ⊢ Fun ◡◡𝐹
Theorem	structfn 17175	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝑀, 𝑁⟩    ⇒   ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))
Theorem	strleun 17176	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩    &   ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩
Theorem	strle1 17177	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    ⇒   ⊢ {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩
Theorem	strle2 17178	Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    ⇒   ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩
Theorem	strle3 17179	Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    &   ⊢ 𝐽 < 𝐾    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝐶 = 𝐾    ⇒   ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾⟩
Theorem	sbcie2s 17180*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ 𝐵 = (𝐹‘𝑊)    &   ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))
Theorem	sbcie3s 17181*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ 𝐵 = (𝐹‘𝑊)    &   ⊢ 𝐶 = (𝐺‘𝑊)    &   ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
7.1.1.2  Substitution of components
Syntax	csts 17182	Set components of a structure.
class sSet
Definition	df-sets 17183*	Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 17252 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 20101, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the +g slot instead of the .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Theorem	reldmsets 17184	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ Rel dom sSet
Theorem	setsvalg 17185	Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
Theorem	setsval 17186	Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
Theorem	fvsetsid 17187	The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)
Theorem	fsets 17188	The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴⟶𝐵)
Theorem	setsdm 17189	The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
Theorem	setsfun 17190	A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))
Theorem	setsfun0 17191	A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 17190 is useful for proofs based on isstruct2 17168 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
Theorem	setsn0fun 17192	The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
Theorem	setsstruct2 17193	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st ‘𝑋), 𝐼, (1st ‘𝑋)), if(𝐼 ≤ (2nd ‘𝑋), (2nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)
Theorem	setsexstruct2 17194*	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → ∃𝑦(𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑦)
Theorem	setsstruct 17195	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)
Theorem	wunsets 17196	Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈)
Theorem	setsres 17197	The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
Theorem	setsabs 17198	Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
Theorem	setscom 17199	Different components can be set in any order. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑊 ∧ 𝐷 ∈ 𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
7.1.1.3  Slots
Syntax	cslot 17200	Extend class notation with the slot function.
class Slot 𝐴