P. 430 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42901-43000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fltnltalem 42901	Lemma for fltnlta 42902. A lower bound for 𝐴 based on pwdif 15791. (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 42902	In a Fermat counterexample, the exponent 𝑁 is less than all three numbers (𝐴, 𝐵, and 𝐶). Note that 𝐴 < 𝐵 (hypothesis) and 𝐵 < 𝐶 (fltltc 42900). See https://youtu.be/EymVXkPWxyc 42900 for an outline. (Contributed by SN, 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)
21.30.9  Exemplar theorems These theorems were added for illustration or pedagogical purposes without the intention of being used, but some may still be moved to main and used, of course.
Theorem	iddii 42903	Version of a1ii 2 with the hypotheses switched. The first hypothesis is redundant so this theorem should not normally appear in a proof. Inference associated with idd 24. (Contributed by SN, 1-Apr-2025.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ 𝜓
Theorem	bicomdALT 42904	Alternate proof of bicomd 223 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 222 depends on bicom1 221 and sylib 218 depends on syl 17). Additionally, the labels bicom1 221 and syl 17 happen to contain fewer characters than bicom 222 and sylib 218. However, neither of these conditions count as a shortening according to conventions 30475. In the first case, the criteria could easily be broken by upstream changes, and in many cases the upstream dependency tree is nontrivial (see orass 921 and pm2.31 922). For the latter case, theorem labels are up to revision, so they are not counted in the size of a proof. (Contributed by SN, 21-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜓))
Theorem	alan 42905	Alias for 19.26 1871 for easier lookup. (Contributed by SN, 12-Aug-2025.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	exor 42906	Alias for 19.43 1883 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	rexor 42907	Alias for r19.43 3104 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	ruvALT 42908	Alternate proof of ruv 9510 with one fewer syntax step thanks to using elirrv 9502 instead of elirr 9504. However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions 30475. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
Theorem	sn-wcdeq 42909	Alternative to wcdeq 3721 and df-cdeq 3722. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3722. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (𝑥 = 𝑦 → 𝜑)
Theorem	sq45 42910	45 squared is 2025. (Contributed by SN, 30-Mar-2025.)
⊢ (;45↑2) = ;;;2025
Theorem	sum9cubes 42911	The sum of the first nine perfect cubes is 2025. (Contributed by SN, 30-Mar-2025.)
⊢ Σ𝑘 ∈ (1...9)(𝑘↑3) = ;;;2025
Theorem	sn-isghm 42912*	Longer proof of isghm 19144, unsuccessfully attempting to simplify isghm 19144 using elovmpo 7603 according to an editorial note (now removed). (Contributed by SN, 7-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
Theorem	aprilfools2025 42913	An abuse of notation. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ {⟨“𝐴𝑝𝑟𝑖𝑙”⟩, ⟨“𝑓𝑜𝑜𝑙𝑠!”⟩} ∈ V
21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12 It is known that ax-10 2146, ax-11 2162, and ax-12 2184 are logically redundant in a weak sense. Practically, they can be replaced with hbn1w 2049, alcomimw 2044, and ax12wlem 2137 as long as you can fully substitute 𝑦 for 𝑥 in the relevant wff (that is, 𝑥 cannot appear in the wff after substituting). This strategy (which I will call a "standard replacement" of axioms) has a lot of potential, for example it works with df-fv 6500 and df-mpt 5180, two very common constructions. But doing a standard replacement of ax-10 2146, ax-11 2162, and ax-12 2184 takes unsatisfyingly long. Usually, if another approach is found, that approach is shorter and better.
Theorem	nfa1w 42914*	Replace ax-10 2146 in nfa1 2156 with a substitution hypothesis. (Contributed by SN, 2-Sep-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	eu6w 42915*	Replace ax-10 2146, ax-12 2184 in eu6 2574 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
Theorem	abbibw 42916*	Replace ax-10 2146, ax-11 2162, ax-12 2184 in abbib 2805 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
Theorem	absnw 42917*	Replace ax-10 2146, ax-11 2162, ax-12 2184 in absn 4600 with a substitution hypothesis. (Contributed by SN, 27-May-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
Theorem	euabsn2w 42918*	Replace ax-10 2146, ax-11 2162, ax-12 2184 in euabsn2 4682 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
21.31  Mathbox for Igor Ieskov
Theorem	cu3addd 42919	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	negexpidd 42920	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 42921*	An extended version of rexlimdvv 3192 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 42922	Lemma for 3cubes 42928. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
Theorem	3cubeslem2 42923	Lemma for 3cubes 42928. Used to show that the denominators in 3cubeslem4 42927 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
Theorem	3cubeslem3l 42924	Lemma for 3cubes 42928. (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 42925	Lemma for 3cubes 42928. (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 42926	Lemma for 3cubes 42928. (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 42927	Lemma for 3cubes 42928. 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 42928*	Every rational number is a sum of three rational cubes. See S. Ryley, The Ladies' Diary 122 (1825), 35. (Contributed by Igor Ieskov, 22-Jan-2024.)
⊢ (𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3)))
21.32  Mathbox for OpenAI
Theorem	rntrclfvOAI 42929	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)
21.33  Mathbox for Stefan O'Rear
21.33.1  Additional elementary logic and set theory
Theorem	moxfr 42930*	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)
21.33.2  Additional theory of functions
Theorem	imaiinfv 42931*	Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵))
21.33.3  Additional topology
Theorem	elrfi 42932*	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 42933*	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 42934*	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 42935*	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 42936*	Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17540), isotone (satisfies mrcss 17539), and idempotent (satisfies mrcidm 17542) 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 42937 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 42937*	Second half of ismrcd1 42936. (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 42938*	A closure function which satisfies sscls 23000, clsidm 23011, cls0 23024, and clsun 36522 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 42939*	A function is a Moore closure operator iff it satisfies mrcssid 17540, mrcss 17539, and mrcidm 17542. (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 42940	Class of Noetherian closure systems.
class NoeACS
Definition	df-nacs 42941*	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 42942*	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 42943*	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 42944	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 42945*	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 42946*	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 42947	A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
Theorem	isnacs3 42948*	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 42949*	Transitivity induction of subsets, lemma for nacsfix 42950. (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 42950*	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 42951	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 42952	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 42953	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 42954	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 42955	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 42956	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 42957	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 42958*	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 42959	Extend class notation to include pre-polynomial rings.
class mzPolyCld
Syntax	cmzp 42960	Extend class notation to include polynomial rings.
class mzPoly
Definition	df-mzpcl 42961*	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 42962. (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 42962	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 42963*	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 42964*	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 42965	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 42962 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (ℤ ↑m (ℤ ↑m 𝑉)) ∈ (mzPolyCld‘𝑉))
Theorem	mzpcln0 42966	Corollary of mzpclall 42965: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) ≠ ∅)
Theorem	mzpcl1 42967	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 42968*	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 42969	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 42970	Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉))
Theorem	dmmzp 42971	mzPoly is defined for all index sets which are sets. This is used with elfvdm 6868 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ dom mzPoly = V
Theorem	mzpincl 42972	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 42973	Constant functions are polynomial. See also mzpconstmpt 42978. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐶}) ∈ (mzPoly‘𝑉))
Theorem	mzpf 42974	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 42975*	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 42976	The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 42979. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉))
Theorem	mzpmul 42977	The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 42980. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉))
Theorem	mzpconstmpt 42978*	A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 42979, mzpmulmpt 42980, mzpnegmpt 42982, mzpsubmpt 42981, mzpexpmpt 42983) 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 42975 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 42979*	Sum of polynomial functions is polynomial. Maps-to version of mzpadd 42976. (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 42980*	Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 42980. (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 42981*	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 42982*	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 42983*	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 42984*	"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 42985	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 42986*	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 42987*	Simplified version of mzpsubst 42986 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 42988*	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 42989*	Lemma for mzpcompact2 42990. (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 42990*	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 42991	coeq0 6214 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 42992	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 42993	Extend class notation to include the family of Diophantine sets.
class Dioph
Definition	df-dioph 42994*	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 16892 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 43001. (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 42995*	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 42996*	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 42997*	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 42998*	Lemma for eldioph2 43000. 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 42999*	Lemma for eldioph2 43000. 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 43000*	Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 42990. (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))