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Theorem List for Metamath Proof Explorer - 42601-42700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-prjspn 42601* Define the n-dimensional projective space function. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Compare df-ehl 25433. This space is considered n-dimensional because the vector space (𝑘 freeLMod (0...𝑛)) is (n+1)-dimensional and the ℙ𝕣𝕠𝕛 function returns equivalence classes with respect to a linear (1-dimensional) relation. (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
 
Theoremprjspnval 42602 Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)
((𝑁 ∈ ℕ0𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
 
Theoremprjspnerlem 42603* A lemma showing that the equivalence relation used in prjspnval2 42604 and the equivalence relation used in prjspval 42589 are equal, but only with the antecedent 𝐾 ∈ DivRing. (Contributed by SN, 15-Jul-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠𝑊)       (𝐾 ∈ DivRing → = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
 
Theoremprjspnval2 42604* Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠𝑊)       ((𝑁 ∈ ℕ0𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ))
 
Theoremprjspner 42605* The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 42601) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 42591 and prjspersym 42593 (see prjspnerlem 42603). Several theorems are covered in one thanks to the theorems around df-er 8743. (Contributed by SN, 14-Aug-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠𝑊)    &   (𝜑𝐾 ∈ DivRing)       (𝜑 Er 𝐵)
 
Theoremprjspnvs 42606* A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 42596 (see prjspnerlem 42603). (Contributed by SN, 8-Aug-2024.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐾)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝐶𝑆)    &   (𝜑𝐶0 )       (𝜑 → (𝐶 · 𝑋) 𝑋)
 
Theoremprjspnssbas 42607 A projective point spans a subset of the (nonzero) affine points. (Contributed by SN, 17-Jan-2025.)
𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ DivRing)       (𝜑𝑃 ⊆ 𝒫 𝐵)
 
Theoremprjspnn0 42608 A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)
𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝐴𝑃)       (𝜑𝐴 ≠ ∅)
 
Theorem0prjspnlem 42609 Lemma for 0prjspn 42614. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑊 = (𝐾 freeLMod (0...0))    &    1 = ((𝐾 unitVec (0...0))‘0)       (𝐾 ∈ DivRing → 1𝐵)
 
Theoremprjspnfv01 42610* Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the value of the zeroth coordinate). (Contributed by SN, 13-Aug-2023.)
𝐹 = (𝑏𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐾)    &    1 = (1r𝐾)    &   𝐼 = (invr𝐾)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝐹𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 ))
 
Theoremprjspner01 42611* Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the equivalence). (Contributed by SN, 13-Aug-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐹 = (𝑏𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &    · = ( ·𝑠𝑊)    &   𝑆 = (Base‘𝐾)    &    0 = (0g𝐾)    &   𝐼 = (invr𝐾)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑𝑋 (𝐹𝑋))
 
Theoremprjspner1 42612* Two vectors whose zeroth coordinate is nonzero are equivalent if and only if they have the same representative in the (n-1)-dimensional affine subspace { x0 = 1 } . For example, vectors in 3D space whose 𝑥 coordinate is nonzero are equivalent iff they intersect at the plane 𝑥 = 1 at the same point (also see section header). (Contributed by SN, 13-Aug-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐹 = (𝑏𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &    · = ( ·𝑠𝑊)    &   𝑆 = (Base‘𝐾)    &    0 = (0g𝐾)    &   𝐼 = (invr𝐾)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋‘0) ≠ 0 )    &   (𝜑 → (𝑌‘0) ≠ 0 )       (𝜑 → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
 
Theorem0prjspnrel 42613* In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &    · = ( ·𝑠𝑊)    &   𝑆 = (Base‘𝐾)    &   𝑊 = (𝐾 freeLMod (0...0))    &    1 = ((𝐾 unitVec (0...0))‘0)       ((𝐾 ∈ DivRing ∧ 𝑋𝐵) → 𝑋 1 )
 
Theorem0prjspn 42614 A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.)
𝑊 = (𝐾 freeLMod (0...0))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})       (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛n𝐾) = {𝐵})
 
Syntaxcprjcrv 42615 Extend class notation with the projective curve function.
class ℙ𝕣𝕠𝕛Crv
 
Definitiondf-prjcrv 42616* Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 42586 and prjspvs 42596). (Contributed by SN, 23-Nov-2024.)
ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g𝑘)}}))
 
Theoremprjcrvfval 42617* Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g𝐾)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ Field)       (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ran 𝐻 ↦ {𝑝𝑃 ∣ ((𝐸𝑓) “ 𝑝) = { 0 }}))
 
Theoremprjcrvval 42618* Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g𝐾)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ Field)    &   (𝜑𝐹 ran 𝐻)       (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝𝑃 ∣ ((𝐸𝐹) “ 𝑝) = { 0 }})
 
Theoremprjcrv0 42619 The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)
𝑌 = ((0...𝑁) mPoly 𝐾)    &    0 = (0g𝑌)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ Field)       (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)
 
21.30.8  Basic reductions for Fermat's Last Theorem
 
Theoremdffltz 42620* Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes (𝑎𝑛) + (𝑏𝑛) = (𝑐𝑛), and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023.)
(∀𝑛 ∈ (ℤ‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥𝑛) + (𝑦𝑛)) ≠ (𝑧𝑛) ↔ ∀𝑛 ∈ (ℤ‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖ {0})((𝑎𝑛) + (𝑏𝑛)) ≠ (𝑐𝑛))
 
Theoremfltmul 42621 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any 𝑁 ∈ ℕ0, so the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.)
(𝜑𝑆 ∈ ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁))
 
Theoremfltdiv 42622 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024.)
(𝜑𝑆 ∈ ℂ)    &   (𝜑𝑆 ≠ 0)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁))
 
Theoremflt0 42623 A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑𝑁 ∈ ℕ)
 
Theoremfltdvdsabdvdsc 42624 Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 42625. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
 
Theoremfltabcoprmex 42625 A counterexample to FLT implies a counterexample to FLT with 𝐴, 𝐵 (assigned to 𝐴 / (𝐴 gcd 𝐵) and 𝐵 / (𝐴 gcd 𝐵)) coprime (by divgcdcoprm0 16698). (Contributed by SN, 20-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → (((𝐴 / (𝐴 gcd 𝐵))↑𝑁) + ((𝐵 / (𝐴 gcd 𝐵))↑𝑁)) = ((𝐶 / (𝐴 gcd 𝐵))↑𝑁))
 
Theoremfltaccoprm 42626 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐴, 𝐶 coprime. (Contributed by SN, 20-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)       (𝜑 → (𝐴 gcd 𝐶) = 1)
 
Theoremfltbccoprm 42627 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 42626 using commutativity of addition. (Contributed by SN, 20-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)       (𝜑 → (𝐵 gcd 𝐶) = 1)
 
Theoremfltabcoprm 42628 A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 42626. (Contributed by SN, 22-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))       (𝜑 → (𝐴 gcd 𝐵) = 1)
 
Theoreminfdesc 42629* Infinite descent. The hypotheses say that 𝑆 is lower bounded, and that if 𝜓 holds for an integer in 𝑆, it holds for a smaller integer in 𝑆. By infinite descent, eventually we cannot go any smaller, therefore 𝜓 holds for no integer in 𝑆. (Contributed by SN, 20-Aug-2024.)
(𝑦 = 𝑥 → (𝜓𝜒))    &   (𝑦 = 𝑧 → (𝜓𝜃))    &   (𝜑𝑆 ⊆ (ℤ𝑀))    &   ((𝜑 ∧ (𝑥𝑆𝜒)) → ∃𝑧𝑆 (𝜃𝑧 < 𝑥))       (𝜑 → {𝑦𝑆𝜓} = ∅)
 
Theoremfltne 42630 If a counterexample to FLT exists, its addends are not equal. (Contributed by SN, 1-Jun-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑𝐴𝐵)
 
Theoremflt4lem 42631 Raising a number to the fourth power is equivalent to squaring it twice. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2))
 
Theoremflt4lem1 42632 Satisfy the antecedent used in several pythagtrip 16867 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))       (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)))
 
Theoremflt4lem2 42633 If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))       (𝜑 → ¬ 2 ∥ 𝐵)
 
Theoremflt4lem3 42634 Equivalent to pythagtriplem4 16852. Show that 𝐶 + 𝐴 and 𝐶𝐴 are coprime. (Contributed by SN, 22-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))       (𝜑 → ((𝐶 + 𝐴) gcd (𝐶𝐴)) = 1)
 
Theoremflt4lem4 42635 If the product of two coprime factors is a perfect square, the factors are perfect squares. (Contributed by SN, 22-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → (𝐴 · 𝐵) = (𝐶↑2))       (𝜑 → (𝐴 = ((𝐴 gcd 𝐶)↑2) ∧ 𝐵 = ((𝐵 gcd 𝐶)↑2)))
 
Theoremflt4lem5 42636 In the context of the lemmas of pythagtrip 16867, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)    &   𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶𝐵))) / 2)       (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)
 
Theoremflt4lem5elem 42637 Version of fltaccoprm 42626 and fltbccoprm 42627 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16760, dvds2addd 16325, and prmdvdsexp 16748, we can show that if two variables are coprime, the third is also coprime to the two. (Contributed by SN, 24-Aug-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑀 = ((𝑅↑2) + (𝑆↑2)))    &   (𝜑 → (𝑅 gcd 𝑆) = 1)       (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))
 
Theoremflt4lem5a 42638 Part 1 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2))
 
Theoremflt4lem5b 42639 Part 2 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2))
 
Theoremflt4lem5c 42640 Part 2 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑𝑁 = (2 · (𝑅 · 𝑆)))
 
Theoremflt4lem5d 42641 Part 3 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑𝑀 = ((𝑅↑2) + (𝑆↑2)))
 
Theoremflt4lem5e 42642 Satisfy the hypotheses of flt4lem4 42635. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → (((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1) ∧ (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)))
 
Theoremflt4lem5f 42643 Final equation of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. Given 𝐴↑4 + 𝐵↑4 = 𝐶↑2, provide a smaller solution. This satisfies the infinite descent condition. (Contributed by SN, 24-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀𝑁))) / 2)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → ((𝑀 gcd (𝐵 / 2))↑2) = (((𝑅 gcd (𝐵 / 2))↑4) + ((𝑆 gcd (𝐵 / 2))↑4)))
 
Theoremflt4lem6 42644 Remove shared factors in a solution to 𝐴↑4 + 𝐵↑4 = 𝐶↑2. (Contributed by SN, 24-Jul-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → (((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐶 / ((𝐴 gcd 𝐵)↑2)) ∈ ℕ) ∧ (((𝐴 / (𝐴 gcd 𝐵))↑4) + ((𝐵 / (𝐴 gcd 𝐵))↑4)) = ((𝐶 / ((𝐴 gcd 𝐵)↑2))↑2)))
 
Theoremflt4lem7 42645* Convert flt4lem5f 42643 into a convenient form for nna4b4nsq 42646. TODO-SN: The change to (𝐴 gcd 𝐵) = 1 points at some inefficiency in the lemmas. (Contributed by SN, 25-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))       (𝜑 → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ) = 1 ∧ ((𝑔↑4) + (↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝐶))
 
Theoremnna4b4nsq 42646 Strengthening of Fermat's last theorem for exponent 4, where the sum is only assumed to be a square. (Contributed by SN, 23-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))
 
Theoremfltltc 42647 (𝐶𝑁) is the largest term and therefore 𝐵 < 𝐶. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑𝐵 < 𝐶)
 
Theoremfltnltalem 42648 Lemma for fltnlta 42649. A lower bound for 𝐴 based on pwdif 15900. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → ((𝐶𝐵) · ((𝐶↑(𝑁 − 1)) + ((𝑁 − 1) · (𝐵↑(𝑁 − 1))))) < (𝐴𝑁))
 
Theoremfltnlta 42649 In a Fermat counterexample, the exponent 𝑁 is less than all three numbers (𝐴, 𝐵, and 𝐶). Note that 𝐴 < 𝐵 (hypothesis) and 𝐵 < 𝐶 (fltltc 42647). See https://youtu.be/EymVXkPWxyc 42647 for an outline. (Contributed by SN, 24-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))    &   (𝜑𝐴 < 𝐵)       (𝜑𝑁 < 𝐴)
 
21.30.9  Exemplar theorems

These theorems were added for illustration or pedagogical purposes without the intention of being used, but some may still be moved to main and used, of course.

 
Theoremiddii 42650 Version of a1ii 2 with the hypotheses switched. The first hypothesis is redundant so this theorem should not normally appear in a proof. Inference associated with idd 24. (Contributed by SN, 1-Apr-2025.) (New usage is discouraged.)
𝜑    &   𝜓       𝜓
 
TheorembicomdALT 42651 Alternate proof of bicomd 223 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 222 depends on bicom1 221 and sylib 218 depends on syl 17). Additionally, the labels bicom1 221 and syl 17 happen to contain fewer characters than bicom 222 and sylib 218. However, neither of these conditions count as a shortening according to conventions 30428. In the first case, the criteria could easily be broken by upstream changes, and in many cases the upstream dependency tree is nontrivial (see orass 921 and pm2.31 922). For the latter case, theorem labels are up to revision, so they are not counted in the size of a proof. (Contributed by SN, 21-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremalan 42652 Alias for 19.26 1867 for easier lookup. (Contributed by SN, 12-Aug-2025.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
 
Theoremexor 42653 Alias for 19.43 1879 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theoremrexor 42654 Alias for r19.43 3119 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
 
TheoremruvALT 42655 Alternate proof of ruv 9639 with one fewer syntax step thanks to using elirrv 9633 instead of elirr 9634. However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions 30428. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
{𝑥𝑥𝑥} = V
 
Theoremsn-wcdeq 42656 Alternative to wcdeq 3771 and df-cdeq 3772. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3772. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (𝑥 = 𝑦𝜑)
 
Theoremsq45 42657 45 squared is 2025. (Contributed by SN, 30-Mar-2025.)
(45↑2) = 2025
 
Theoremsum9cubes 42658 The sum of the first nine perfect cubes is 2025. (Contributed by SN, 30-Mar-2025.)
Σ𝑘 ∈ (1...9)(𝑘↑3) = 2025
 
Theoremsn-isghm 42659* Longer proof of isghm 19245, unsuccessfully attempting to simplify isghm 19245 using elovmpo 7677 according to an editorial note (now removed). (Contributed by SN, 7-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
 
Theoremaprilfools2025 42660 An abuse of notation. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
{⟨“𝐴𝑝𝑟𝑖𝑙”⟩, ⟨“𝑓𝑜𝑜𝑙𝑠!”⟩} ∈ V
 
21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12

It is known that ax-10 2138, ax-11 2154, and ax-12 2174 are logically redundant in a weak sense. Practically, they can be replaced with hbn1w 2043, alcomimw 2039, and ax12wlem 2129 as long as you can fully substitute 𝑦 for 𝑥 in the relevant wff (that is, 𝑥 cannot appear in the wff after substituting).

This strategy (which I will call a "standard replacement" of axioms) has a lot of potential, for example it works with df-fv 6570 and df-mpt 5231, two very common constructions. But doing a standard replacement of ax-10 2138, ax-11 2154, and ax-12 2174 takes unsatisfyingly long. Usually, if another approach is found, that approach is shorter and better.

 
Theoremnfa1w 42661* Replace ax-10 2138 in nfa1 2148 with a substitution hypothesis. (Contributed by SN, 2-Sep-2025.)
(𝑥 = 𝑦 → (𝜑𝜓))       𝑥𝑥𝜑
 
Theoremeu6w 42662* Replace ax-10 2138, ax-12 2174 in eu6 2571 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜃))       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
Theoremabbibw 42663* Replace ax-10 2138, ax-11 2154, ax-12 2174 in abbib 2808 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝑦 → (𝜓𝜒))       ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
 
Theoremabsnw 42664* Replace ax-10 2138, ax-11 2154, ax-12 2174 in absn 4649 with a substitution hypothesis. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑𝜓))       ({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
 
Theoremeuabsn2w 42665* Replace ax-10 2138, ax-11 2154, ax-12 2174 in euabsn2 4729 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑧 → (𝜑𝜃))       (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
 
Theoremsn-tz6.12-2 42666* tz6.12-2 6894 without ax-10 2138, ax-11 2154, ax-12 2174. Improves 118 theorems. (Contributed by SN, 27-May-2025.)
(¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
 
21.31  Mathbox for Igor Ieskov
 
Theoremcu3addd 42667 Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (((𝐴 + 𝐵) + 𝐶)↑3) = (((((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))) + (((3 · ((𝐴↑2) · 𝐶)) + (((3 · 2) · (𝐴 · 𝐵)) · 𝐶)) + (3 · ((𝐵↑2) · 𝐶)))) + (((3 · (𝐴 · (𝐶↑2))) + (3 · (𝐵 · (𝐶↑2)))) + (𝐶↑3))))
 
Theoremsqnegd 42668 The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (-𝐴↑2) = (𝐴↑2))
 
Theoremnegexpidd 42669 The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ¬ 2 ∥ 𝑁)       (𝜑 → ((𝐴𝑁) + (-𝐴𝑁)) = 0)
 
Theoremrexlimdv3d 42670* An extended version of rexlimdvv 3209 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜓𝜒))
 
Theorem3cubeslem1 42671 Lemma for 3cubes 42677. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
 
Theorem3cubeslem2 42672 Lemma for 3cubes 42677. Used to show that the denominators in 3cubeslem4 42676 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
 
Theorem3cubeslem3l 42673 Lemma for 3cubes 42677. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
Theorem3cubeslem3r 42674 Lemma for 3cubes 42677. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
Theorem3cubeslem3 42675 Lemma for 3cubes 42677. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)))
 
Theorem3cubeslem4 42676 Lemma for 3cubes 42677. This is Ryley's explicit formula for decomposing a rational 𝐴 into a sum of three rational cubes. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 = (((((((3↑3) · (𝐴↑3)) − 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3) + ((((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)) + (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)))
 
Theorem3cubes 42677* Every rational number is a sum of three rational cubes. See S. Ryley, The Ladies' Diary 122 (1825), 35. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3)))
 
21.32  Mathbox for OpenAI
 
TheoremrntrclfvOAI 42678 The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
 
21.33  Mathbox for Stefan O'Rear
 
21.33.1  Additional elementary logic and set theory
 
Theoremmoxfr 42679* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
21.33.2  Additional theory of functions
 
Theoremimaiinfv 42680* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
 
21.33.3  Additional topology
 
Theoremelrfi 42681* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
 
Theoremelrfirn 42682* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
 
Theoremelrfirn2 42683* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉 ∧ ∀𝑦𝐼 𝐶𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦𝐼𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 𝐶)))
 
Theoremcmpfiiin 42684* In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   ((𝜑𝑘𝐼) → 𝑆 ∈ (Clsd‘𝐽))    &   ((𝜑 ∧ (𝑙𝐼𝑙 ∈ Fin)) → (𝑋 𝑘𝑙 𝑆) ≠ ∅)       (𝜑 → (𝑋 𝑘𝐼 𝑆) ≠ ∅)
 
21.33.4  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 42685* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17661), isotone (satisfies mrcss 17660), and idempotent (satisfies mrcidm 17663) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 42686 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
 
Theoremismrcd2 42686* Second half of ismrcd1 42685. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
 
Theoremistopclsd 42687* A closure function which satisfies sscls 23079, clsidm 23090, cls0 23103, and clsun 36310 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))    &   (𝜑 → (𝐹‘∅) = ∅)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))    &   𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)}       (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
 
Theoremismrc 42688* A function is a Moore closure operator iff it satisfies mrcssid 17661, mrcss 17660, and mrcidm 17663. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
 
21.33.5  Algebraic closure systems
 
Syntaxcnacs 42689 Class of Noetherian closure systems.
class NoeACS
 
Definitiondf-nacs 42690* Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
 
Theoremisnacs 42691* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
 
Theoremnacsfg 42692* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
 
Theoremisnacs2 42693 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
 
Theoremmrefg2 42694* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
 
Theoremmrefg3 42695* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
 
Theoremnacsacs 42696 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
 
Theoremisnacs3 42697* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
 
Theoremincssnn0 42698* Transitivity induction of subsets, lemma for nacsfix 42699. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0𝐵 ∈ (ℤ𝐴)) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremnacsfix 42699* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))
 
21.33.6  Miscellanea 1. Map utilities
 
Theoremconstmap 42700 A constant (represented without dummy variables) is an element of a function set.

Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets. (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

𝐴 ∈ V    &   𝐶 ∈ V       (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶m 𝐴))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49035
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