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Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsn-subf 41301 subf 11462 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
− :(ℂ × ℂ)⟶ℂ
 
Theoremresubeqsub 41302 Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝐴𝐵))
 
Theoremsubresre 41303 Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
= ( − ↾ (ℝ × ℝ))
 
Theoremaddinvcom 41304 A number commutes with its additive inverse. Compare remulinvcom 41305. (Contributed by SN, 5-May-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 0)       (𝜑 → (𝐵 + 𝐴) = 0)
 
Theoremremulinvcom 41305 A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11174. (Contributed by SN, 5-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 1)       (𝜑 → (𝐵 · 𝐴) = 1)
 
Theoremremullid 41306 Commuted version of ax-1rid 11180 without ax-mulcom 11174. (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
 
Theoremsn-1ticom 41307 Lemma for sn-mullid 41308 and it1ei 41309. (Contributed by SN, 27-May-2024.)
(1 · i) = (i · 1)
 
Theoremsn-mullid 41308 mullid 11213 without ax-mulcom 11174. (Contributed by SN, 27-May-2024.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
 
Theoremit1ei 41309 1 is a multiplicative identity for i (see sn-mullid 41308 for commuted version). (Contributed by SN, 1-Jun-2024.)
(i · 1) = i
 
Theoremipiiie0 41310 The multiplicative inverse of i (per i4 14168) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
(i + (i · (i · i))) = 0
 
Theoremremulcand 41311 Commuted version of remulcan2d 41177 without ax-mulcom 11174. (Contributed by SN, 21-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremsn-0tie0 41312 Lemma for sn-mul02 41313. Commuted version of sn-it0e0 41288. (Contributed by SN, 30-Jun-2024.)
(0 · i) = 0
 
Theoremsn-mul02 41313 mul02 11392 without ax-mulcom 11174. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11174 for an outline. (Contributed by SN, 30-Jun-2024.)
(𝐴 ∈ ℂ → (0 · 𝐴) = 0)
 
Theoremsn-ltaddpos 41314 ltaddpos 11704 without ax-mulcom 11174. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremsn-ltaddneg 41315 ltaddneg 11429 without ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
 
Theoremreposdif 41316 Comparison of two numbers whose difference is positive. Compare posdif 11707. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 𝐴)))
 
Theoremrelt0neg1 41317 Comparison of a real and its negative to zero. Compare lt0neg1 11720. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 − 𝐴)))
 
Theoremrelt0neg2 41318 Comparison of a real and its negative to zero. Compare lt0neg2 11721. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 − 𝐴) < 0))
 
Theoremsn-addlt0d 41319 The sum of negative numbers is negative. (Contributed by SN, 25-Jan-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 0)    &   (𝜑𝐵 < 0)       (𝜑 → (𝐴 + 𝐵) < 0)
 
Theoremsn-addgt0d 41320 The sum of positive numbers is positive. Proof of addgt0d 11789 without ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremsn-nnne0 41321 nnne0 12246 without ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)
 
Theoremreelznn0nn 41322 elznn0nn 12572 restated using df-resub 41239. (Contributed by SN, 25-Jan-2025.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 − 𝑁) ∈ ℕ)))
 
Theoremnn0addcom 41323 Addition is commutative for nonnegative integers. Proven without ax-mulcom 11174. (Contributed by SN, 1-Feb-2025.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremzaddcomlem 41324 Lemma for zaddcom 41325. (Contributed by SN, 1-Feb-2025.)
(((𝐴 ∈ ℝ ∧ (0 − 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremzaddcom 41325 Addition is commutative for integers. Proven without ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremrenegmulnnass 41326 Move multiplication by a natural number inside and outside negation. (Contributed by SN, 25-Jan-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((0 − 𝐴) · 𝑁) = (0 − (𝐴 · 𝑁)))
 
Theoremnn0mulcom 41327 Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremzmulcomlem 41328 Lemma for zmulcom 41329. (Contributed by SN, 25-Jan-2025.)
(((𝐴 ∈ ℝ ∧ (0 − 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremzmulcom 41329 Multiplication is commutative for integers. Proven without ax-mulcom 11174. From this result and grpcominv1 41082, we can show that rationals commute under multiplication without using ax-mulcom 11174. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremmulgt0con1dlem 41330 Lemma for mulgt0con1d 41331. Contraposes a positive deduction to a negative deduction. (Contributed by SN, 26-Jun-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (0 < 𝐴 → 0 < 𝐵))    &   (𝜑 → (𝐴 = 0 → 𝐵 = 0))       (𝜑 → (𝐵 < 0 → 𝐴 < 0))
 
Theoremmulgt0con1d 41331 Counterpart to mulgt0con2d 41332, though not a lemma of anything. This is the first use of ax-pre-mulgt0 11187. (Contributed by SN, 26-Jun-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐵)    &   (𝜑 → (𝐴 · 𝐵) < 0)       (𝜑𝐴 < 0)
 
Theoremmulgt0con2d 41332 Lemma for mulgt0b2d 41333 and contrapositive of mulgt0 11291. (Contributed by SN, 26-Jun-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → (𝐴 · 𝐵) < 0)       (𝜑𝐵 < 0)
 
Theoremmulgt0b2d 41333 Biconditional, deductive form of mulgt0 11291. The second factor is positive iff the product is. Note that the commuted form cannot be proven since resubdi 41269 does not have a commuted form. (Contributed by SN, 26-Jun-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑 → (0 < 𝐵 ↔ 0 < (𝐴 · 𝐵)))
 
Theoremsn-ltmul2d 41334 ltmul2d 13058 without ax-mulcom 11174. (Contributed by SN, 26-Jun-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 < 𝐶)       (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 𝐴 < 𝐵))
 
Theoremsn-0lt1 41335 0lt1 11736 without ax-mulcom 11174. (Contributed by SN, 13-Feb-2024.)
0 < 1
 
Theoremsn-ltp1 41336 ltp1 12054 without ax-mulcom 11174. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
Theoremreneg1lt0 41337 Lemma for sn-inelr 41338. (Contributed by SN, 1-Jun-2024.)
(0 − 1) < 0
 
Theoremsn-inelr 41338 inelr 12202 without ax-mulcom 11174. (Contributed by SN, 1-Jun-2024.)
¬ i ∈ ℝ
 
Theoremitrere 41339 i times a real is real iff the real is zero. (Contributed by SN, 27-Jun-2024.)
(𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
 
Theoremretire 41340 Commuted version of itrere 41339. (Contributed by SN, 27-Jun-2024.)
(𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
 
Theoremcnreeu 41341 The reals in the expression given by cnre 11211 uniquely define a complex number. (Contributed by SN, 27-Jun-2024.)
(𝜑𝑟 ∈ ℝ)    &   (𝜑𝑠 ∈ ℝ)    &   (𝜑𝑡 ∈ ℝ)    &   (𝜑𝑢 ∈ ℝ)       (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡𝑠 = 𝑢)))
 
Theoremsn-sup2 41342* sup2 12170 with exactly the same proof except for using sn-ltp1 41336 instead of ltp1 12054, saving ax-mulcom 11174. (Contributed by SN, 26-Jun-2024.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 (𝑦 < 𝑥𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
21.28.6  Projective spaces

Looking at a corner in 3D space, one can see three right angles. It is impossible to draw three lines in 2D space such that any two of these lines are perpendicular, but a good enough representation is made by casting lines from the 2D surface. Points along the same cast line are collapsed into one point on the 2D surface.

In many cases, the 2D surface is smaller than whatever needs to be represented. If the lines cast were perpendicular to the 2D surface, then only areas as small as the 2D surface could be represented. To fix this, the lines need to get further apart as they go farther from the 2D surface. On the other side of the 2D surface the lines will get closer together and intersect at a point (because it's defined that way).

From this perspective, two parallel lines in 3D space will be represented by two lines that seem to intersect at a point "at infinity". Considering all maximal classes of parallel lines on a 2D plane in 3D space, these classes will all appear to intersect at different points at infinity, forming a line at infinity. Therefore the real projective plane can be thought of as the real affine plane together with the line at infinity.

The projective plane takes care of some exceptions that may be found in the affine plane. For example, consider the curve that is the zeroes of 𝑦 = 𝑥↑2. Any line connecting the point (0, 1) to the x-axis intersects with the curve twice, except for the vertical line between (0, 1) and (0, 0). In the projective plane, the curve becomes an ellipse and there is no exception.

While it may not seem like it, points at infinity and points corresponding to the affine plane are the same type of point. Consider a line going through the origin in 3D (affine) space. Either it intersects the plane 𝑧 = 1 once, or it is entirely within the plane 𝑧 = 0. If it is entirely within the plane 𝑧 = 0, then it corresponds to the point at infinity intersecting all lines on the plane 𝑧 = 1 with the same slope. Else it corresponds to the point in the 2D plane 𝑧 = 1 that it intersects. So there is a bijection between 3D lines through the origin and points on the real projective plane.

The concept of projective spaces generalizes the projective plane to any dimension.

 
Syntaxcprjsp 41343 Extend class notation with the projective space function.
class ℙ𝕣𝕠𝕛
 
Definitiondf-prjsp 41344* Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, making equivalent rational multiples of real numbers). Compare df-lsatoms 37846. (Contributed by BJ and SN, 29-Apr-2023.)
ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
 
Theoremprjspval 41345* Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)
𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &    · = ( ·𝑠𝑉)    &   𝑆 = (Scalar‘𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
 
Theoremprjsprel 41346* Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}       (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
 
Theoremprjspertr 41347* The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       ((𝑉 ∈ LMod ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
 
Theoremprjsperref 41348* The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LMod → (𝑋𝐵𝑋 𝑋))
 
Theoremprjspersym 41349* The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       ((𝑉 ∈ LVec ∧ 𝑋 𝑌) → 𝑌 𝑋)
 
Theoremprjsper 41350* The relation used to define ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LVec → Er 𝐵)
 
Theoremprjspreln0 41351* Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)       (𝑉 ∈ LVec → (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌))))
 
Theoremprjspvs 41352* A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑉 ∈ LVec ∧ 𝑋𝐵𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) 𝑋)
 
Theoremprjsprellsp 41353* Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &   𝑁 = (LSpan‘𝑉)       ((𝑉 ∈ LVec ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
 
Theoremprjspeclsp 41354* The vectors equivalent to a vector 𝑋 are the nonzero vectors in the span of 𝑋. (Contributed by Steven Nguyen, 6-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &   𝑁 = (LSpan‘𝑉)       ((𝑉 ∈ LVec ∧ 𝑋𝐵) → [𝑋] = ((𝑁‘{𝑋}) ∖ {(0g𝑉)}))
 
Theoremprjspval2 41355* Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.)
0 = (0g𝑉)    &   𝐵 = ((Base‘𝑉) ∖ { 0 })    &   𝑁 = (LSpan‘𝑉)       (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = 𝑧𝐵 {((𝑁‘{𝑧}) ∖ { 0 })})
 
Syntaxcprjspn 41356 Extend class notation with the n-dimensional projective space function.
class ℙ𝕣𝕠𝕛n
 
Definitiondf-prjspn 41357* 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 24903. 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 41358 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 41359* A lemma showing that the equivalence relation used in prjspnval2 41360 and the equivalence relation used in prjspval 41345 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 41360* 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 41361* The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 41357) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 41347 and prjspersym 41349 (see prjspnerlem 41359). Several theorems are covered in one thanks to the theorems around df-er 8703. (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 41362* A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 41352 (see prjspnerlem 41359). (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 41363 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 41364 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 41365 Lemma for 0prjspn 41370. 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 41366* 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 41367* 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 41368* 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 41369* 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 41370 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 41371 Extend class notation with the projective curve function.
class ℙ𝕣𝕠𝕛Crv
 
Definitiondf-prjcrv 41372* 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 41172 and prjspvs 41352). (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 41373* 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 41374* 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 41375 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)
 
21.28.7  Basic reductions for Fermat's Last Theorem
 
Theoremdffltz 41376* 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 41377 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 41378 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 41379 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 41380 Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 41381. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
 
Theoremfltabcoprmex 41381 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 16602). (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 41382 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 41383 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 41382 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 41384 A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 41382. (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 41385* 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 41386 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 41387 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 41388 Satisfy the antecedent used in several pythagtrip 16767 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 41389 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 41390 Equivalent to pythagtriplem4 16752. 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 41391 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 41392 In the context of the lemmas of pythagtrip 16767, 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 41393 Version of fltaccoprm 41382 and fltbccoprm 41383 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16663, dvds2addd 16235, and prmdvdsexp 16652, 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 41394 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 41395 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 41396 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 41397 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 41398 Satisfy the hypotheses of flt4lem4 41391. (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 41399 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 41400 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)))
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