P. 429 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)

Type	Label	Description
Statement
21.33.3  Additional topology
Theorem	elrfi 42801*	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 42802*	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 42803*	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 42804*	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) ≠ ∅)
21.33.4  Characterization of closure operators. Kuratowski closure axioms
Theorem	ismrcd1 42805*	Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17533), isotone (satisfies mrcss 17532), and idempotent (satisfies mrcidm 17535) 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 42806 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 42806*	Second half of ismrcd1 42805. (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 42807*	A closure function which satisfies sscls 22981, clsidm 22992, cls0 23005, and clsun 36383 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 42808*	A function is a Moore closure operator iff it satisfies mrcssid 17533, mrcss 17532, and mrcidm 17535. (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)))))
21.33.5  Algebraic closure systems
Syntax	cnacs 42809	Class of Noetherian closure systems.
class NoeACS
Definition	df-nacs 42810*	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 42811*	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 42812*	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 42813	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 42814*	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 42815*	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 42816	A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
Theorem	isnacs3 42817*	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 42818*	Transitivity induction of subsets, lemma for nacsfix 42819. (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 42819*	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))
21.33.6  Miscellanea 1. Map utilities
Theorem	constmap 42820	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 42821	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 42822	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 42823	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 42824	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 42825	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 42826	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)
21.33.7  Miscellanea for polynomials
Theorem	mptfcl 42827*	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))
21.33.8  Multivariate polynomials over the integers
Syntax	cmzpcl 42828	Extend class notation to include pre-polynomial rings.
class mzPolyCld
Syntax	cmzp 42829	Extend class notation to include polynomial rings.
class mzPoly
Definition	df-mzpcl 42830*	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 42831. (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))})
Definition	df-mzp 42831	Polynomials over ℤ with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
Theorem	mzpclval 42832*	Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPolyCld‘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))})
Theorem	elmzpcl 42833*	Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (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)))))
Theorem	mzpclall 42834	The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 42831 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (ℤ ↑m (ℤ ↑m 𝑉)) ∈ (mzPolyCld‘𝑉))
Theorem	mzpcln0 42835	Corollary of mzpclall 42834: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) ≠ ∅)
Theorem	mzpcl1 42836	Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃)
Theorem	mzpcl2 42837*	Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)
Theorem	mzpcl34 42838	Defining properties 3 and 4 of a polynomially closed function set 𝑃: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))
Theorem	mzpval 42839	Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉))
Theorem	dmmzp 42840	mzPoly is defined for all index sets which are sets. This is used with elfvdm 6865 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ dom mzPoly = V
Theorem	mzpincl 42841	Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉))
Theorem	mzpconst 42842	Constant functions are polynomial. See also mzpconstmpt 42847. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐶}) ∈ (mzPoly‘𝑉))
Theorem	mzpf 42843	A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ (𝐹 ∈ (mzPoly‘𝑉) → 𝐹:(ℤ ↑m 𝑉)⟶ℤ)
Theorem	mzpproj 42844*	A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑉 ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑋)) ∈ (mzPoly‘𝑉))
Theorem	mzpadd 42845	The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 42848. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉))
Theorem	mzpmul 42846	The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 42849. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉))
Theorem	mzpconstmpt 42847*	A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 42848, mzpmulmpt 42849, mzpnegmpt 42851, mzpsubmpt 42850, mzpexpmpt 42852) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 42844 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐶) ∈ (mzPoly‘𝑉))
Theorem	mzpaddmpt 42848*	Sum of polynomial functions is polynomial. Maps-to version of mzpadd 42845. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 + 𝐵)) ∈ (mzPoly‘𝑉))
Theorem	mzpmulmpt 42849*	Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 42849. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 · 𝐵)) ∈ (mzPoly‘𝑉))
Theorem	mzpsubmpt 42850*	The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘𝑉))
Theorem	mzpnegmpt 42851*	Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ ((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ -𝐴) ∈ (mzPoly‘𝑉))
Theorem	mzpexpmpt 42852*	Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴↑𝐷)) ∈ (mzPoly‘𝑉))
Theorem	mzpindd 42853*	"Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑉) → 𝜃)    &   ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜁)    &   ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜎)    &   ⊢ (𝑥 = ((ℤ ↑m 𝑉) × {𝑓}) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))    &   ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))    &   ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝜌)
Theorem	mzpmfp 42854	Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)
⊢ (mzPoly‘𝐼) = ran (𝐼 eval ℤring)
Theorem	mzpsubst 42855*	Substituting polynomials for the variables of a polynomial results in a polynomial. 𝐺 is expected to depend on 𝑦 and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
Theorem	mzprename 42856*	Simplified version of mzpsubst 42855 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ 𝑅:𝑉⟶𝑊) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ 𝑅))) ∈ (mzPoly‘𝑊))
Theorem	mzpresrename 42857*	A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) ∈ (mzPoly‘𝑊))
Theorem	mzpcompact2lem 42858*	Lemma for mzpcompact2 42859. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))
Theorem	mzpcompact2 42859*	Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))
21.33.9  Miscellanea for Diophantine sets 1
Theorem	coeq0i 42860	coeq0 6211 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅)
Theorem	fzsplit1nn0 42861	Split a finite 1-based set of integers in the middle, allowing either end to be empty ((1...0)). (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
21.33.10  Diophantine sets 1: definitions
Syntax	cdioph 42862	Extend class notation to include the family of Diophantine sets.
class Dioph
Definition	df-dioph 42863*	A Diophantine set is a set of positive integers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes ℤ (via mzPoly) and ℕ0 (to define the zero sets); the former could be avoided by considering coincidence sets of ℕ0 polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 16886 that implicitly restricting variables to ℕ0 adds no expressive power over allowing them to range over ℤ. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 42870. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)}))
Theorem	eldiophb 42864*	Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ (𝐷 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
Theorem	eldioph 42865*	Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))
Theorem	diophrw 42866*	Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)
⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)} = {𝑎 ∣ ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)})
Theorem	eldioph2lem1 42867*	Lemma for eldioph2 42869. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
Theorem	eldioph2lem2 42868*	Lemma for eldioph2 42869. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
Theorem	eldioph2 42869*	Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 42859. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))
Theorem	eldioph2b 42870*	While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 ∖ (1...𝑁)). For instance, in diophin 42879 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
Theorem	eldiophelnn0 42871	Remove antecedent on 𝐵 from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐵 ∈ ℕ0)
Theorem	eldioph3b 42872*	Define Diophantine sets in terms of polynomials with variables indexed by ℕ. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 42864 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
Theorem	eldioph3 42873*	Inference version of eldioph3b 42872 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))
21.33.11  Diophantine sets 2 miscellanea
Theorem	ellz1 42874	Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ (𝐵 ∈ ℤ → (𝐴 ∈ (ℤ ∖ (ℤ≥‘(𝐵 + 1))) ↔ (𝐴 ∈ ℤ ∧ 𝐴 ≤ 𝐵)))
Theorem	lzunuz 42875	The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≤ (𝐴 + 1)) → ((ℤ ∖ (ℤ≥‘(𝐴 + 1))) ∪ (ℤ≥‘𝐵)) = ℤ)
Theorem	fz1eqin 42876	Express a one-based finite range as the intersection of lower integers with ℕ. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ))
Theorem	lzenom 42877	Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝑁 ∈ ℤ → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ≈ ω)
Theorem	elmapresaunres2 42878	fresaunres2 6703 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra
Theorem	diophin 42879	If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁))
Theorem	diophun 42880	If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))
Theorem	eldiophss 42881	Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵)))
21.33.13  Diophantine sets 3: construction
Theorem	diophrex 42882*	Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
Theorem	eq0rabdioph 42883*	This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))
Theorem	eqrabdioph 42884*	Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))
Theorem	0dioph 42885	The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ ℕ0 → ∅ ∈ (Dioph‘𝐴))
Theorem	vdioph 42886	The "universal" set (as large as possible given eldiophss 42881) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ ℕ0 → (ℕ0 ↑m (1...𝐴)) ∈ (Dioph‘𝐴))
Theorem	anrabdioph 42887*	Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓)} ∈ (Dioph‘𝑁))
Theorem	orrabdioph 42888*	Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓)} ∈ (Dioph‘𝑁))
Theorem	3anrabdioph 42889*	Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓 ∧ 𝜒)} ∈ (Dioph‘𝑁))
Theorem	3orrabdioph 42890*	Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)} ∈ (Dioph‘𝑁))
21.33.14  Diophantine sets 4 miscellanea
Theorem	2sbcrex 42891*	Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Theorem	sbcrexgOLD 42892*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3823 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
Theorem	2sbcrexOLD 42893*	Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7399 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Theorem	sbc2rex 42894*	Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑)
Theorem	sbc2rexgOLD 42895*	Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7399 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))
Theorem	sbc4rex 42896*	Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑)
Theorem	sbc4rexgOLD 42897*	Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7399 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))
Theorem	sbcrot3 42898*	Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
Theorem	sbcrot5 42899*	Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
Theorem	sbccomieg 42900*	Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)