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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | numexp1 16401 | Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝐴↑1) = 𝐴 | ||
Theorem | numexpp1 16402 | Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (𝑀 + 1) = 𝑁 & ⊢ ((𝐴↑𝑀) · 𝐴) = 𝐶 ⇒ ⊢ (𝐴↑𝑁) = 𝐶 | ||
Theorem | numexp2x 16403 | Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (2 · 𝑀) = 𝑁 & ⊢ (𝐴↑𝑀) = 𝐷 & ⊢ (𝐷 · 𝐷) = 𝐶 ⇒ ⊢ (𝐴↑𝑁) = 𝐶 | ||
Theorem | decsplit0b 16404 | Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((𝐴 · (;10↑0)) + 𝐵) = (𝐴 + 𝐵) | ||
Theorem | decsplit0 16405 | Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((𝐴 · (;10↑0)) + 0) = 𝐴 | ||
Theorem | decsplit1 16406 | 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 16407 | 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 16408 | 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 12151. (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 16409 | Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ (2↑4) = ;16 | ||
Theorem | 2exp6 16410 | 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 | 2exp8 16411 | Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ (2↑8) = ;;256 | ||
Theorem | 2exp16 16412 | Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ (2↑;16) = ;;;;65536 | ||
Theorem | 3exp3 16413 | Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ (3↑3) = ;27 | ||
Theorem | 2expltfac 16414 | The factorial grows faster than two to the power 𝑁. (Contributed by Mario Carneiro, 15-Sep-2016.) |
⊢ (𝑁 ∈ (ℤ≥‘4) → (2↑𝑁) < (!‘𝑁)) | ||
Theorem | cshwsidrepsw 16415 | 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 16416 | 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 16417* | 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 16418* | 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 16419* | 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 16420* | 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 16421* | 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 16422* | 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 16423* | 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 16424* | 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 16425* | 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 16426* | 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)) | ||
Theorem | prmlem0 16427* | Lemma for prmlem1 16429 and prmlem2 16441. (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 16428* | 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 16429 | 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 16430 | 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ 5 ∈ ℙ | ||
Theorem | 6nprm 16431 | 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ ¬ 6 ∈ ℙ | ||
Theorem | 7prm 16432 | 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ 7 ∈ ℙ | ||
Theorem | 8nprm 16433 | 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ ¬ 8 ∈ ℙ | ||
Theorem | 9nprm 16434 | 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ ¬ 9 ∈ ℙ | ||
Theorem | 10nprm 16435 | 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ ¬ ;10 ∈ ℙ | ||
Theorem | 11prm 16436 | 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ ;11 ∈ ℙ | ||
Theorem | 13prm 16437 | 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ ;13 ∈ ℙ | ||
Theorem | 17prm 16438 | 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ ;17 ∈ ℙ | ||
Theorem | 19prm 16439 | 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ ;19 ∈ ℙ | ||
Theorem | 23prm 16440 | 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
⊢ ;23 ∈ ℙ | ||
Theorem | prmlem2 16441 |
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 us 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 16457).
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 16442 | 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;37 ∈ ℙ | ||
Theorem | 43prm 16443 | 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;43 ∈ ℙ | ||
Theorem | 83prm 16444 | 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;83 ∈ ℙ | ||
Theorem | 139prm 16445 | 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;;139 ∈ ℙ | ||
Theorem | 163prm 16446 | 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;;163 ∈ ℙ | ||
Theorem | 317prm 16447 | 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;;317 ∈ ℙ | ||
Theorem | 631prm 16448 | 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
⊢ ;;631 ∈ ℙ | ||
Theorem | prmo4 16449 | The primorial of 4. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘4) = 6 | ||
Theorem | prmo5 16450 | The primorial of 5. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘5) = ;30 | ||
Theorem | prmo6 16451 | The primorial of 6. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘6) = ;30 | ||
Theorem | 1259lem1 16452 | Lemma for 1259prm 16457. 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 16453 | Lemma for 1259prm 16457. 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 16454 | Lemma for 1259prm 16457. 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 16455 | Lemma for 1259prm 16457. 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 16456 | Lemma for 1259prm 16457. 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 16457 | 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 16458 | Lemma for 2503prm 16461. 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 16459 | Lemma for 2503prm 16461. 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 16460 | Lemma for 2503prm 16461. 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 16461 | 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 16462 | Lemma for 4001prm 16466. 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 16463 | Lemma for 4001prm 16466. 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 16464 | Lemma for 4001prm 16466. 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 16465 | Lemma for 4001prm 16466. 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 16466 | 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 | ||
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 16497 and strfv 16519. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 16497, 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〉}. 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 16479. 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 20546 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 16709 addresses a different situation. It is not possible to have SubGroup 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 16709 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 16845 and df-acs 16848. SubGroup 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 SubGroup 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. | ||
Syntax | cstr 16467 | Extend class notation with the class of structures with components numbered below 𝐴. |
class Struct | ||
Syntax | cnx 16468 | Extend class notation with the structure component index extractor. |
class ndx | ||
Syntax | csts 16469 | Set components of a structure. |
class sSet | ||
Syntax | cslot 16470 | Extend class notation with the slot function. |
class Slot 𝐴 | ||
Syntax | cbs 16471 | Extend class notation with the class of all base set extractors. |
class Base | ||
Syntax | cress 16472 | Extend class notation with the extensible structure builder restriction operator. |
class ↾s | ||
Definition | df-struct 16473* |
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 6366, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16483: 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 4818). This is used critically in strle1 16580, strle2 16581, strle3 16582 and strleun 16579 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16631 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 16632, 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 20485. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ Struct = {〈𝑓, 𝑥〉 ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))} | ||
Definition | df-ndx 16474 | Define the structure component index extractor. See theorem ndxarg 16496 to understand its purpose. The restriction to ℕ ensures that ndx is a set. The restriction to some set is necessary since I is a proper class. In principle, we could have chosen ℂ or (if we revise all structure component definitions such as df-base 16477) another set such as the set of finite ordinals ω (df-om 7570). (Contributed by NM, 4-Sep-2011.) |
⊢ ndx = ( I ↾ ℕ) | ||
Definition | df-slot 16475* |
Define the slot extractor for extensible structures. The class
Slot 𝐴 is a function whose argument can be
any set, although it is
meaningful only if that set is a member of an extensible structure (such
as a partially ordered set (df-poset 17544) or a group (df-grp 18044)).
Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴‘𝑆) is defined to be (𝑆‘𝐴), where 𝐴 will typically be a small nonzero natural number. Each extensible structure 𝑆 is a function defined on specific natural number "slots", and this function extracts the value at a particular slot. The special "structure" ndx, defined as the identity function restricted to ℕ, can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 16496). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression (Base‘ndx) in theorems and proofs instead of its value 1). The class Slot cannot be defined as (𝑥 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓‘𝑥))) because each Slot 𝐴 is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 6676). It is necessary to allow proper classes as values of Slot 𝐴 since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴)) | ||
Theorem | sloteq 16476 | Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.) |
⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | ||
Definition | df-base 16477 | Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ Base = Slot 1 | ||
Definition | df-sets 16478* | 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 16479 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 19169, 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})) | ||
Definition | df-ress 16479* |
Define a multifunction restriction operator for extensible structures,
which 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).
(Credit for this operator goes to Mario Carneiro.) See ressbas 16542 for the altered base set, and resslem 16545 (subrg0 19471, ressplusg 16600, subrg1 19474, ressmulr 16613) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) | ||
Theorem | brstruct 16480 | The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ Rel Struct | ||
Theorem | isstruct2 16481 | 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 16482 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V) | ||
Theorem | structn0fun 16483 | A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | ||
Theorem | isstruct 16484 | 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 16485 | Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) | ||
Theorem | structfung 16486 | The converse of the converse of a structure is a function. Closed form of structfun 16487. (Contributed by AV, 12-Nov-2021.) |
⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹) | ||
Theorem | structfun 16487 | 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 16488 | 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 | slotfn 16489 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝐸 = Slot 𝑁 ⇒ ⊢ 𝐸 Fn V | ||
Theorem | strfvnd 16490 | Deduction version of strfvn 16493. (Contributed by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) | ||
Theorem | basfn 16491 | The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
⊢ Base Fn V | ||
Theorem | wunndx 16492 | Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) ⇒ ⊢ (𝜑 → ndx ∈ 𝑈) | ||
Theorem | strfvn 16493 |
Value of a structure component extractor 𝐸. Normally, 𝐸 is a
defined constant symbol such as Base (df-base 16477) and 𝑁 is a
fixed integer such as 1. 𝑆 is a structure, i.e. a
specific
member of a class of structures such as Poset
(df-poset 17544) where
𝑆
∈ Poset.
Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 16519. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.) |
⊢ 𝑆 ∈ V & ⊢ 𝐸 = Slot 𝑁 ⇒ ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) | ||
Theorem | strfvss 16494 | A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐸 = Slot 𝑁 ⇒ ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 | ||
Theorem | wunstr 16495 | Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) | ||
Theorem | ndxarg 16496 | Get the numeric argument from a defined structure component extractor such as df-base 16477. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐸‘ndx) = 𝑁 | ||
Theorem | ndxid 16497 |
A structure component extractor is defined by its own index. This
theorem, together with strfv 16519 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 16477 and the ;10 in
df-ple 16573, making it easier to change should the need
arise.
For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), 𝐿〉} rather than {〈1, 𝐵〉, 〈;10, 𝐿〉}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐸 = Slot (𝐸‘ndx) | ||
Theorem | strndxid 16498 | The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) | ||
Theorem | reldmsets 16499 | The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ Rel dom sSet | ||
Theorem | setsvalg 16500 | 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})) |
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