P. 402 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-prjsp 40101*	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, equivocating rational multiples of real numbers). (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
Theorem	prjspval 40102*	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) ∖ {(0g‘𝑉)})    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))
Theorem	prjsprel 40103*	Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    ⇒   ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))
Theorem	prjspertr 40104*	The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LMod ∧ (𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍)) → 𝑋 ∼ 𝑍)
Theorem	prjsperref 40105*	The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋))
Theorem	prjspersym 40106*	The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∼ 𝑌) → 𝑌 ∼ 𝑋)
Theorem	prjsper 40107*	The relation used to define ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → ∼ Er 𝐵)
Theorem	prjspreln0 40108*	Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌))))
Theorem	prjspvs 40109*	A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋)
Theorem	prjsprellsp 40110*	Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∼ 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
Theorem	prjspeclsp 40111*	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⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵) → [𝑋] ∼ = ((𝑁‘{𝑋}) ∖ {(0g‘𝑉)}))
Theorem	prjspval2 40112*	Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ 0 = (0g‘𝑉)    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ { 0 })    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = ∪ 𝑧 ∈ 𝐵 {((𝑁‘{𝑧}) ∖ { 0 })})
Syntax	cprjspn 40113	Extend class notation with the n-dimensional projective space function.
class ℙ𝕣𝕠𝕛n
Definition	df-prjspn 40114*	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 24255. 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))))
Theorem	prjspnval 40115	Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
Theorem	prjspnerlem 40116*	A lemma showing that the equivalence relation used in prjspnval2 40117 and the equivalence relation used in prjspval 40102 are equal, but only with the antecedent 𝐾 ∈ DivRing. (Contributed by SN, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
Theorem	prjspnval2 40117*	Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ ))
Theorem	prjspner 40118*	The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 40114) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 40104 and prjspersym 40106 (see prjspnerlem 40116). Several theorems are covered in one thanks to the theorems around df-er 8380. (Contributed by SN, 14-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    ⇒   ⊢ (𝜑 → ∼ Er 𝐵)
Theorem	prjspnvs 40119*	A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 40109 (see prjspnerlem 40116). (Contributed by SN, 8-Aug-2024.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋)
Theorem	0prjspnlem 40120	Lemma for 0prjspn 40125. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ (𝐾 ∈ DivRing → 1 ∈ 𝐵)
Theorem	prjspnfv01 40121*	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 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 1 = (1r‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 ))
Theorem	prjspner01 40122*	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⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))
Theorem	prjspner1 40123*	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⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋‘0) ≠ 0 )    &   ⊢ (𝜑 → (𝑌‘0) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
Theorem	0prjspnrel 40124*	In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 )
Theorem	0prjspn 40125	A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    ⇒   ⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛n𝐾) = {𝐵})
20.26.8  Basic reductions for Fermat's Last Theorem
Theorem	dffltz 40126*	Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes (𝑎↑𝑛) + (𝑏↑𝑛) = (𝑐↑𝑛), 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 ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ∀𝑛 ∈ (ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))
Theorem	fltmul 40127	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, hence the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁))
Theorem	fltdiv 40128	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.)
⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑆 ≠ 0)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁))
Theorem	flt0 40129	A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ)
Theorem	fltdvdsabdvdsc 40130	Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 40131. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
Theorem	fltabcoprmex 40131	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 16203). (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝐴 / (𝐴 gcd 𝐵))↑𝑁) + ((𝐵 / (𝐴 gcd 𝐵))↑𝑁)) = ((𝐶 / (𝐴 gcd 𝐵))↑𝑁))
Theorem	fltaccoprm 40132	A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐴, 𝐶 coprime. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)
Theorem	fltbccoprm 40133	A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 40132 using commutativity of addition. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 gcd 𝐶) = 1)
Theorem	fltabcoprm 40134	A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 40132. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)
Theorem	infdesc 40135*	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 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑆 ⊆ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝜒)) → ∃𝑧 ∈ 𝑆 (𝜃 ∧ 𝑧 < 𝑥))    ⇒   ⊢ (𝜑 → {𝑦 ∈ 𝑆 ∣ 𝜓} = ∅)
Theorem	fltne 40136	If a counterexample to FLT exists, its addends are not equal. (Contributed by SN, 1-Jun-2023.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	flt4lem 40137	Raising a number to the fourth power is equivalent to squaring it twice. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2))
Theorem	flt4lem1 40138	Satisfy the antecedent used in several pythagtrip 16368 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ 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)))
Theorem	flt4lem2 40139	If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
Theorem	flt4lem3 40140	Equivalent to pythagtriplem4 16353. Show that 𝐶 + 𝐴 and 𝐶 − 𝐴 are coprime. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1)
Theorem	flt4lem4 40141	If the product of two coprime factors is a perfect square, the factors are perfect squares. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (𝐴 = ((𝐴 gcd 𝐶)↑2) ∧ 𝐵 = ((𝐵 gcd 𝐶)↑2)))
Theorem	flt4lem5 40142	In the context of the lemmas of pythagtrip 16368, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)
⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)    &   ⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)    ⇒   ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)
Theorem	flt4lem5elem 40143	Version of fltaccoprm 40132 and fltbccoprm 40133 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16264, dvds2addd 15834, and prmdvdsexp 16253, we can show that if two variables are coprime, the third is also coprime to the two. (Contributed by SN, 24-Aug-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))    &   ⊢ (𝜑 → (𝑅 gcd 𝑆) = 1)    ⇒   ⊢ (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))
Theorem	flt4lem5a 40144	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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2))
Theorem	flt4lem5b 40145	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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2))
Theorem	flt4lem5c 40146	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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆)))
Theorem	flt4lem5d 40147	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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))
Theorem	flt4lem5e 40148	Satisfy the hypotheses of flt4lem4 40141. EDITORIAL: This is not minimized! (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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ 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)))
Theorem	flt4lem5f 40149	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)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ 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)))
Theorem	flt4lem6 40150	Remove shared factors in a solution to 𝐴↑4 + 𝐵↑4 = 𝐶↑2. (Contributed by SN, 24-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ 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)))
Theorem	flt4lem7 40151*	Convert flt4lem5f 40149 into a convenient form for nna4b4nsq 40152. TODO-SN: The change to (𝐴 gcd 𝐵) = 1 points at some inefficiency in the lemmas. EDITORIAL: This is not minimized! (Contributed by SN, 25-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝐶))
Theorem	nna4b4nsq 40152	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.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))
Theorem	fltltc 40153	(𝐶↑𝑁) is the largest term and therefore 𝐵 < 𝐶. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → 𝐵 < 𝐶)
Theorem	fltnltalem 40154	Lemma for fltnlta 40155. A lower bound for 𝐴 based on pwdif 15413. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → ((𝐶 − 𝐵) · ((𝐶↑(𝑁 − 1)) + ((𝑁 − 1) · (𝐵↑(𝑁 − 1))))) < (𝐴↑𝑁))
Theorem	fltnlta 40155	𝑁 is less than 𝐴. See https://www.youtu.be/EymVXkPWxyc for an outline. (Contributed by Steven Nguyen, 24-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝑁 < 𝐴)
20.27  Mathbox for Igor Ieskov
Theorem	binom2d 40156	Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
Theorem	cu3addd 40157	Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 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))))
Theorem	sqnegd 40158	The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (-𝐴↑2) = (𝐴↑2))
Theorem	negexpidd 40159	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0)
Theorem	rexlimdv3d 40160*	An extended version of rexlimdvv 3205 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))
Theorem	3cubeslem1 40161	Lemma for 3cubes 40167. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
Theorem	3cubeslem2 40162	Lemma for 3cubes 40167. Used to show that the denominators in 3cubeslem4 40166 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
Theorem	3cubeslem3l 40163	Lemma for 3cubes 40167. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ 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)))))))))
Theorem	3cubeslem3r 40164	Lemma for 3cubes 40167. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ 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)))))))))
Theorem	3cubeslem3 40165	Lemma for 3cubes 40167. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ 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)))
Theorem	3cubeslem4 40166	Lemma for 3cubes 40167. 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.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → 𝐴 = (((((((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)))
Theorem	3cubes 40167*	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)))
20.28  Mathbox for OpenAI
Theorem	rntrclfvOAI 40168	The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)
20.29  Mathbox for Stefan O'Rear
20.29.1  Additional elementary logic and set theory
Theorem	moxfr 40169*	Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ ∃!𝑦 𝑥 = 𝐴    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
20.29.2  Additional theory of functions
Theorem	imaiinfv 40170*	Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))
20.29.3  Additional topology
Theorem	elrfi 40171*	Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑣)))
Theorem	elrfirn 40172*	Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
Theorem	elrfirn2 40173*	Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))
Theorem	cmpfiiin 40174*	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    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽))    &   ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅)
20.29.4  Characterization of closure operators. Kuratowski closure axioms
Theorem	ismrcd1 40175*	Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17092), isotone (satisfies mrcss 17091), and idempotent (satisfies mrcidm 17094) 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 40176 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
Theorem	ismrcd2 40176*	Second half of ismrcd1 40175. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
Theorem	istopclsd 40177*	A closure function which satisfies sscls 21925, clsidm 21936, cls0 21949, and clsun 34211 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))    &   ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)}    ⇒   ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
Theorem	ismrc 40178*	A function is a Moore closure operator iff it satisfies mrcssid 17092, mrcss 17091, and mrcidm 17094. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))
20.29.5  Algebraic closure systems
Syntax	cnacs 40179	Class of Noetherian closure systems.
class NoeACS
Definition	df-nacs 40180*	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) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
Theorem	isnacs 40181*	Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
Theorem	nacsfg 40182*	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) ∧ 𝑆 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔))
Theorem	isnacs2 40183	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))
Theorem	mrefg2 40184*	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 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔)))
Theorem	mrefg3 40185*	Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔)))
Theorem	nacsacs 40186	A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
Theorem	isnacs3 40187*	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 → ∪ 𝑠 ∈ 𝑠)))
Theorem	incssnn0 40188*	Transitivity induction of subsets, lemma for nacsfix 40189. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))
Theorem	nacsfix 40189*	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:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦))
20.29.6  Miscellanea 1. Map utilities
Theorem	constmap 40190	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    ⇒   ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴))
Theorem	mapco2g 40191	Renaming indices in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸))
Theorem	mapco2 40192	Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ 𝐸 ∈ V    ⇒   ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸))
Theorem	mapfzcons 40193	Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ 𝑀 = (𝑁 + 1)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ (𝐵 ↑m (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (𝐴 ∪ {⟨𝑀, 𝐶⟩}) ∈ (𝐵 ↑m (1...𝑀)))
Theorem	mapfzcons1 40194	Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ 𝑀 = (𝑁 + 1)    ⇒   ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩}) ↾ (1...𝑁)) = 𝐴)
Theorem	mapfzcons1cl 40195	A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ 𝑀 = (𝑁 + 1)    ⇒   ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑀)) → (𝐴 ↾ (1...𝑁)) ∈ (𝐵 ↑m (1...𝑁)))
Theorem	mapfzcons2 40196	Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
⊢ 𝑀 = (𝑁 + 1)    ⇒   ⊢ ((𝐴 ∈ (𝐵 ↑m (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩})‘𝑀) = 𝐶)
20.29.7  Miscellanea for polynomials
Theorem	mptfcl 40197*	Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ ((𝑡 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶 → (𝑡 ∈ 𝐴 → 𝐵 ∈ 𝐶))
20.29.8  Multivariate polynomials over the integers
Syntax	cmzpcl 40198	Extend class notation to include pre-polynomial rings.
class mzPolyCld
Syntax	cmzp 40199	Extend class notation to include polynomial rings.
class mzPoly
Definition	df-mzpcl 40200*	Define the polynomially closed function rings over an arbitrary index set 𝑣. The set (mzPolyCld‘𝑣) contains all sets of functions from (ℤ ↑m 𝑣) to ℤ which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself (mzPoly‘𝑣); see df-mzp 40201. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑣 (𝑥 ∈ (ℤ ↑m 𝑣) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))})