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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremflt4lem7 41401* Convert flt4lem5f 41399 into a convenient form for nna4b4nsq 41402. TODO-SN: The change to (𝐴 gcd 𝐡) = 1 points at some inefficiency in the lemmas. (Contributed by SN, 25-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝐴)    &   (πœ‘ β†’ (𝐴 gcd 𝐡) = 1)    &   (πœ‘ β†’ ((𝐴↑4) + (𝐡↑4)) = (𝐢↑2))    β‡’   (πœ‘ β†’ βˆƒπ‘™ ∈ β„• (βˆƒπ‘” ∈ β„• βˆƒβ„Ž ∈ β„• (Β¬ 2 βˆ₯ 𝑔 ∧ ((𝑔 gcd β„Ž) = 1 ∧ ((𝑔↑4) + (β„Žβ†‘4)) = (𝑙↑2))) ∧ 𝑙 < 𝐢))
 
Theoremnna4b4nsq 41402 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.)
(πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ β„•)    β‡’   (πœ‘ β†’ ((𝐴↑4) + (𝐡↑4)) β‰  (𝐢↑2))
 
Theoremfltltc 41403 (𝐢↑𝑁) is the largest term and therefore 𝐡 < 𝐢. (Contributed by Steven Nguyen, 22-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ ((𝐴↑𝑁) + (𝐡↑𝑁)) = (𝐢↑𝑁))    β‡’   (πœ‘ β†’ 𝐡 < 𝐢)
 
Theoremfltnltalem 41404 Lemma for fltnlta 41405. A lower bound for 𝐴 based on pwdif 15814. (Contributed by Steven Nguyen, 22-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ ((𝐴↑𝑁) + (𝐡↑𝑁)) = (𝐢↑𝑁))    β‡’   (πœ‘ β†’ ((𝐢 βˆ’ 𝐡) Β· ((𝐢↑(𝑁 βˆ’ 1)) + ((𝑁 βˆ’ 1) Β· (𝐡↑(𝑁 βˆ’ 1))))) < (𝐴↑𝑁))
 
Theoremfltnlta 41405 In a Fermat counterexample, the exponent 𝑁 is less than all three numbers (𝐴, 𝐡, and 𝐢). Note that 𝐴 < 𝐡 (hypothesis) and 𝐡 < 𝐢 (fltltc 41403). See https://youtu.be/EymVXkPWxyc 41403 for an outline. (Contributed by SN, 24-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ ((𝐴↑𝑁) + (𝐡↑𝑁)) = (𝐢↑𝑁))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ 𝑁 < 𝐴)
 
21.28.8  Exemplar theorems

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

 
Theoremiddii 41406 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.)
πœ‘    &   πœ“    β‡’   πœ“
 
TheorembicomdALT 41407 Alternate proof of bicomd 222 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 221 depends on bicom1 220 and sylib 217 depends on syl 17). Additionally, the labels bicom1 220 and syl 17 happen to contain fewer characters than bicom 221 and sylib 217. However, neither of these conditions count as a shortening according to conventions 29653. 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.)
(πœ‘ β†’ (πœ“ ↔ πœ’))    β‡’   (πœ‘ β†’ (πœ’ ↔ πœ“))
 
Theoremelabgw 41408* Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on π‘₯ and 𝐴. This is to elabg 3667 what sbievw2 2100 is to sbievw 2096. (Contributed by SN, 20-Apr-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝐴 β†’ (πœ“ ↔ πœ’))    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ {π‘₯ ∣ πœ‘} ↔ πœ’))
 
Theoremelab2gw 41409* Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on π‘₯ and 𝐴, which is not usually significant since 𝐡 is usually a constant. (Contributed by SN, 16-May-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝐴 β†’ (πœ“ ↔ πœ’))    &   π΅ = {π‘₯ ∣ πœ‘}    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ 𝐡 ↔ πœ’))
 
Theoremelrab2w 41410* Membership in a restricted class abstraction. This is to elrab2 3687 what elab2gw 41409 is to elab2g 3671. (Contributed by SN, 2-Sep-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝐴 β†’ (πœ“ ↔ πœ’))    &   πΆ = {π‘₯ ∈ 𝐡 ∣ πœ‘}    β‡’   (𝐴 ∈ 𝐢 ↔ (𝐴 ∈ 𝐡 ∧ πœ’))
 
TheoremruvALT 41411 Alternate proof of ruv 9597 with one fewer syntax step thanks to using elirrv 9591 instead of elirr 9592. 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 29653. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
{π‘₯ ∣ π‘₯ βˆ‰ π‘₯} = V
 
Theoremsn-wcdeq 41412 Alternative to wcdeq 3760 and df-cdeq 3761. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3761. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (π‘₯ = 𝑦 β†’ πœ‘)
 
Theoremsq45 41413 45 squared is 2025. (Contributed by SN, 30-Mar-2025.)
(45↑2) = 2025
 
Theoremsum9cubes 41414 The sum of the first nine perfect cubes is 2025. (Contributed by SN, 30-Mar-2025.)
Ξ£π‘˜ ∈ (1...9)(π‘˜β†‘3) = 2025
 
Theoremacos1half 41415 The arccosine of 1 / 2 is Ο€ / 3. (Contributed by SN, 31-Aug-2024.)
(arccosβ€˜(1 / 2)) = (Ο€ / 3)
 
Theoremaprilfools2025 41416 An abuse of notation. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
{βŸ¨β€œπ΄π‘π‘Ÿπ‘–π‘™β€βŸ©, βŸ¨β€œπ‘“π‘œπ‘œπ‘™π‘ !β€βŸ©} ∈ V
 
21.29  Mathbox for Igor Ieskov
 
Theorembinom2d 41417 Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐡)↑2) = (((𝐴↑2) + (2 Β· (𝐴 Β· 𝐡))) + (𝐡↑2)))
 
Theoremcu3addd 41418 Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (((𝐴 + 𝐡) + 𝐢)↑3) = (((((𝐴↑3) + (3 Β· ((𝐴↑2) Β· 𝐡))) + ((3 Β· (𝐴 Β· (𝐡↑2))) + (𝐡↑3))) + (((3 Β· ((𝐴↑2) Β· 𝐢)) + (((3 Β· 2) Β· (𝐴 Β· 𝐡)) Β· 𝐢)) + (3 Β· ((𝐡↑2) Β· 𝐢)))) + (((3 Β· (𝐴 Β· (𝐢↑2))) + (3 Β· (𝐡 Β· (𝐢↑2)))) + (𝐢↑3))))
 
Theoremsqnegd 41419 The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (-𝐴↑2) = (𝐴↑2))
 
Theoremnegexpidd 41420 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑁)    β‡’   (πœ‘ β†’ ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0)
 
Theoremrexlimdv3d 41421* An extended version of rexlimdvv 3211 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
(πœ‘ β†’ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐢) β†’ (πœ“ β†’ πœ’)))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐢 πœ“ β†’ πœ’))
 
Theorem3cubeslem1 41422 Lemma for 3cubes 41428. (Contributed by Igor Ieskov, 22-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ 0 < (((𝐴 + 1)↑2) βˆ’ 𝐴))
 
Theorem3cubeslem2 41423 Lemma for 3cubes 41428. Used to show that the denominators in 3cubeslem4 41427 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ Β¬ ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3) = 0)
 
Theorem3cubeslem3l 41424 Lemma for 3cubes 41428. (Contributed by Igor Ieskov, 22-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ (𝐴 Β· (((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3)↑3)) = (((𝐴↑7) Β· (3↑9)) + (((𝐴↑6) Β· (3↑9)) + (((𝐴↑5) Β· ((3↑8) + (3↑8))) + (((𝐴↑4) Β· (((3↑7) Β· 2) + (3↑6))) + (((𝐴↑3) Β· ((3↑6) + (3↑6))) + (((𝐴↑2) Β· (3↑5)) + (𝐴 Β· (3↑3)))))))))
 
Theorem3cubeslem3r 41425 Lemma for 3cubes 41428. (Contributed by Igor Ieskov, 22-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ ((((((3↑3) Β· (𝐴↑3)) βˆ’ 1)↑3) + (((-((3↑3) Β· (𝐴↑3)) + ((3↑2) Β· 𝐴)) + 1)↑3)) + ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴))↑3)) = (((𝐴↑7) Β· (3↑9)) + (((𝐴↑6) Β· (3↑9)) + (((𝐴↑5) Β· ((3↑8) + (3↑8))) + (((𝐴↑4) Β· (((3↑7) Β· 2) + (3↑6))) + (((𝐴↑3) Β· ((3↑6) + (3↑6))) + (((𝐴↑2) Β· (3↑5)) + (𝐴 Β· (3↑3)))))))))
 
Theorem3cubeslem3 41426 Lemma for 3cubes 41428. (Contributed by Igor Ieskov, 22-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ (𝐴 Β· (((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3)↑3)) = ((((((3↑3) Β· (𝐴↑3)) βˆ’ 1)↑3) + (((-((3↑3) Β· (𝐴↑3)) + ((3↑2) Β· 𝐴)) + 1)↑3)) + ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴))↑3)))
 
Theorem3cubeslem4 41427 Lemma for 3cubes 41428. 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.)
(πœ‘ β†’ 𝐴 ∈ β„š)    β‡’   (πœ‘ β†’ 𝐴 = (((((((3↑3) Β· (𝐴↑3)) βˆ’ 1) / ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3))↑3) + ((((-((3↑3) Β· (𝐴↑3)) + ((3↑2) Β· 𝐴)) + 1) / ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3))↑3)) + (((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) / ((((3↑3) Β· (𝐴↑2)) + ((3↑2) Β· 𝐴)) + 3))↑3)))
 
Theorem3cubes 41428* 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 ∈ β„š ↔ βˆƒπ‘Ž ∈ β„š βˆƒπ‘ ∈ β„š βˆƒπ‘ ∈ β„š 𝐴 = (((π‘Žβ†‘3) + (𝑏↑3)) + (𝑐↑3)))
 
21.30  Mathbox for OpenAI
 
TheoremrntrclfvOAI 41429 The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
(𝑅 ∈ 𝑉 β†’ ran (t+β€˜π‘…) = ran 𝑅)
 
21.31  Mathbox for Stefan O'Rear
 
21.31.1  Additional elementary logic and set theory
 
Theoremmoxfr 41430* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
𝐴 ∈ V    &   βˆƒ!𝑦 π‘₯ = 𝐴    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒ*π‘₯πœ‘ ↔ βˆƒ*π‘¦πœ“)
 
21.31.2  Additional theory of functions
 
Theoremimaiinfv 41431* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐹 Fn 𝐴 ∧ 𝐡 βŠ† 𝐴) β†’ ∩ π‘₯ ∈ 𝐡 (πΉβ€˜π‘₯) = ∩ (𝐹 β€œ 𝐡))
 
21.31.3  Additional topology
 
Theoremelrfi 41432* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐡 ∈ 𝑉 ∧ 𝐢 βŠ† 𝒫 𝐡) β†’ (𝐴 ∈ (fiβ€˜({𝐡} βˆͺ 𝐢)) ↔ βˆƒπ‘£ ∈ (𝒫 𝐢 ∩ Fin)𝐴 = (𝐡 ∩ ∩ 𝑣)))
 
Theoremelrfirn 41433* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐡 ∈ 𝑉 ∧ 𝐹:πΌβŸΆπ’« 𝐡) β†’ (𝐴 ∈ (fiβ€˜({𝐡} βˆͺ ran 𝐹)) ↔ βˆƒπ‘£ ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐡 ∩ ∩ 𝑦 ∈ 𝑣 (πΉβ€˜π‘¦))))
 
Theoremelrfirn2 41434* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐡 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝐼 𝐢 βŠ† 𝐡) β†’ (𝐴 ∈ (fiβ€˜({𝐡} βˆͺ ran (𝑦 ∈ 𝐼 ↦ 𝐢))) ↔ βˆƒπ‘£ ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐡 ∩ ∩ 𝑦 ∈ 𝑣 𝐢)))
 
Theoremcmpfiiin 41435* 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 = βˆͺ 𝐽    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ 𝑆 ∈ (Clsdβ€˜π½))    &   ((πœ‘ ∧ (𝑙 βŠ† 𝐼 ∧ 𝑙 ∈ Fin)) β†’ (𝑋 ∩ ∩ π‘˜ ∈ 𝑙 𝑆) β‰  βˆ…)    β‡’   (πœ‘ β†’ (𝑋 ∩ ∩ π‘˜ ∈ 𝐼 𝑆) β‰  βˆ…)
 
21.31.4  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 41436* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17561), isotone (satisfies mrcss 17560), and idempotent (satisfies mrcidm 17563) 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 41437 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ dom (𝐹 ∩ I ) ∈ (Mooreβ€˜π΅))
 
Theoremismrcd2 41437* Second half of ismrcd1 41436. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ 𝐹 = (mrClsβ€˜dom (𝐹 ∩ I )))
 
Theoremistopclsd 41438* A closure function which satisfies sscls 22560, clsidm 22571, cls0 22584, and clsun 35213 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝒫 π΅βŸΆπ’« 𝐡)    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ π‘₯ βŠ† (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡) β†’ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯))    &   (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)    &   ((πœ‘ ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† 𝐡) β†’ (πΉβ€˜(π‘₯ βˆͺ 𝑦)) = ((πΉβ€˜π‘₯) βˆͺ (πΉβ€˜π‘¦)))    &   π½ = {𝑧 ∈ 𝒫 𝐡 ∣ (πΉβ€˜(𝐡 βˆ– 𝑧)) = (𝐡 βˆ– 𝑧)}    β‡’   (πœ‘ β†’ (𝐽 ∈ (TopOnβ€˜π΅) ∧ (clsβ€˜π½) = 𝐹))
 
Theoremismrc 41439* A function is a Moore closure operator iff it satisfies mrcssid 17561, mrcss 17560, and mrcidm 17563. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 ∈ (mrCls β€œ (Mooreβ€˜π΅)) ↔ (𝐡 ∈ V ∧ 𝐹:𝒫 π΅βŸΆπ’« 𝐡 ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝐡 ∧ 𝑦 βŠ† π‘₯) β†’ (π‘₯ βŠ† (πΉβ€˜π‘₯) ∧ (πΉβ€˜π‘¦) βŠ† (πΉβ€˜π‘₯) ∧ (πΉβ€˜(πΉβ€˜π‘₯)) = (πΉβ€˜π‘₯)))))
 
21.31.5  Algebraic closure systems
 
Syntaxcnacs 41440 Class of Noetherian closure systems.
class NoeACS
 
Definitiondf-nacs 41441* 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 = (π‘₯ ∈ V ↦ {𝑐 ∈ (ACSβ€˜π‘₯) ∣ βˆ€π‘  ∈ 𝑐 βˆƒπ‘” ∈ (𝒫 π‘₯ ∩ Fin)𝑠 = ((mrClsβ€˜π‘)β€˜π‘”)})
 
Theoremisnacs 41442* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrClsβ€˜πΆ)    β‡’   (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”)))
 
Theoremnacsfg 41443* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrClsβ€˜πΆ)    β‡’   ((𝐢 ∈ (NoeACSβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”))
 
Theoremisnacs2 41444 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrClsβ€˜πΆ)    β‡’   (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
 
Theoremmrefg2 41445* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrClsβ€˜πΆ)    β‡’   (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
 
Theoremmrefg3 41446* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrClsβ€˜πΆ)    β‡’   ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 βŠ† (πΉβ€˜π‘”)))
 
Theoremnacsacs 41447 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐢 ∈ (NoeACSβ€˜π‘‹) β†’ 𝐢 ∈ (ACSβ€˜π‘‹))
 
Theoremisnacs3 41448* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (Mooreβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝒫 𝐢((toIncβ€˜π‘ ) ∈ Dirset β†’ βˆͺ 𝑠 ∈ 𝑠)))
 
Theoremincssnn0 41449* Transitivity induction of subsets, lemma for nacsfix 41450. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((βˆ€π‘₯ ∈ β„•0 (πΉβ€˜π‘₯) βŠ† (πΉβ€˜(π‘₯ + 1)) ∧ 𝐴 ∈ β„•0 ∧ 𝐡 ∈ (β„€β‰₯β€˜π΄)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))
 
Theoremnacsfix 41450* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((𝐢 ∈ (NoeACSβ€˜π‘‹) ∧ 𝐹:β„•0⟢𝐢 ∧ βˆ€π‘₯ ∈ β„•0 (πΉβ€˜π‘₯) βŠ† (πΉβ€˜(π‘₯ + 1))) β†’ βˆƒπ‘¦ ∈ β„•0 βˆ€π‘§ ∈ (β„€β‰₯β€˜π‘¦)(πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
 
21.31.6  Miscellanea 1. Map utilities
 
Theoremconstmap 41451 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    &   πΆ ∈ V    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴 Γ— {𝐡}) ∈ (𝐢 ↑m 𝐴))
 
Theoremmapco2g 41452 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 ∈ (𝐡 ↑m 𝐢) ∧ 𝐷:𝐸⟢𝐢) β†’ (𝐴 ∘ 𝐷) ∈ (𝐡 ↑m 𝐸))
 
Theoremmapco2 41453 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    β‡’   ((𝐴 ∈ (𝐡 ↑m 𝐢) ∧ 𝐷:𝐸⟢𝐢) β†’ (𝐴 ∘ 𝐷) ∈ (𝐡 ↑m 𝐸))
 
Theoremmapfzcons 41454 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)    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐴 ∈ (𝐡 ↑m (1...𝑁)) ∧ 𝐢 ∈ 𝐡) β†’ (𝐴 βˆͺ {βŸ¨π‘€, 𝐢⟩}) ∈ (𝐡 ↑m (1...𝑀)))
 
Theoremmapfzcons1 41455 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)    β‡’   (𝐴 ∈ (𝐡 ↑m (1...𝑁)) β†’ ((𝐴 βˆͺ {βŸ¨π‘€, 𝐢⟩}) β†Ύ (1...𝑁)) = 𝐴)
 
Theoremmapfzcons1cl 41456 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)    β‡’   (𝐴 ∈ (𝐡 ↑m (1...𝑀)) β†’ (𝐴 β†Ύ (1...𝑁)) ∈ (𝐡 ↑m (1...𝑁)))
 
Theoremmapfzcons2 41457 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)    β‡’   ((𝐴 ∈ (𝐡 ↑m (1...𝑁)) ∧ 𝐢 ∈ 𝐡) β†’ ((𝐴 βˆͺ {βŸ¨π‘€, 𝐢⟩})β€˜π‘€) = 𝐢)
 
21.31.7  Miscellanea for polynomials
 
Theoremmptfcl 41458* Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑑 ∈ 𝐴 ↦ 𝐡):𝐴⟢𝐢 β†’ (𝑑 ∈ 𝐴 β†’ 𝐡 ∈ 𝐢))
 
21.31.8  Multivariate polynomials over the integers
 
Syntaxcmzpcl 41459 Extend class notation to include pre-polynomial rings.
class mzPolyCld
 
Syntaxcmzp 41460 Extend class notation to include polynomial rings.
class mzPoly
 
Definitiondf-mzpcl 41461* Define the polynomially closed function rings over an arbitrary index set 𝑣. The set (mzPolyCldβ€˜π‘£) contains all sets of functions from (β„€ ↑m 𝑣) to β„€ 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 (mzPolyβ€˜π‘£); see df-mzp 41462. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (β„€ ↑m (β„€ ↑m 𝑣)) ∣ ((βˆ€π‘– ∈ β„€ ((β„€ ↑m 𝑣) Γ— {𝑖}) ∈ 𝑝 ∧ βˆ€π‘— ∈ 𝑣 (π‘₯ ∈ (β„€ ↑m 𝑣) ↦ (π‘₯β€˜π‘—)) ∈ 𝑝) ∧ βˆ€π‘“ ∈ 𝑝 βˆ€π‘” ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f Β· 𝑔) ∈ 𝑝))})
 
Definitiondf-mzp 41462 Polynomials over β„€ 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 ↦ ∩ (mzPolyCldβ€˜π‘£))
 
Theoremmzpclval 41463* Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝑉 ∈ V β†’ (mzPolyCldβ€˜π‘‰) = {𝑝 ∈ 𝒫 (β„€ ↑m (β„€ ↑m 𝑉)) ∣ ((βˆ€π‘– ∈ β„€ ((β„€ ↑m 𝑉) Γ— {𝑖}) ∈ 𝑝 ∧ βˆ€π‘— ∈ 𝑉 (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (π‘₯β€˜π‘—)) ∈ 𝑝) ∧ βˆ€π‘“ ∈ 𝑝 βˆ€π‘” ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f Β· 𝑔) ∈ 𝑝))})
 
Theoremelmzpcl 41464* Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝑉 ∈ V β†’ (𝑃 ∈ (mzPolyCldβ€˜π‘‰) ↔ (𝑃 βŠ† (β„€ ↑m (β„€ ↑m 𝑉)) ∧ ((βˆ€π‘– ∈ β„€ ((β„€ ↑m 𝑉) Γ— {𝑖}) ∈ 𝑃 ∧ βˆ€π‘— ∈ 𝑉 (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (π‘₯β€˜π‘—)) ∈ 𝑃) ∧ βˆ€π‘“ ∈ 𝑃 βˆ€π‘” ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f Β· 𝑔) ∈ 𝑃)))))
 
Theoremmzpclall 41465 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 41462 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝑉 ∈ V β†’ (β„€ ↑m (β„€ ↑m 𝑉)) ∈ (mzPolyCldβ€˜π‘‰))
 
Theoremmzpcln0 41466 Corollary of mzpclall 41465: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝑉 ∈ V β†’ (mzPolyCldβ€˜π‘‰) β‰  βˆ…)
 
Theoremmzpcl1 41467 Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝑃 ∈ (mzPolyCldβ€˜π‘‰) ∧ 𝐹 ∈ β„€) β†’ ((β„€ ↑m 𝑉) Γ— {𝐹}) ∈ 𝑃)
 
Theoremmzpcl2 41468* Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝑃 ∈ (mzPolyCldβ€˜π‘‰) ∧ 𝐹 ∈ 𝑉) β†’ (𝑔 ∈ (β„€ ↑m 𝑉) ↦ (π‘”β€˜πΉ)) ∈ 𝑃)
 
Theoremmzpcl34 41469 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 ∈ (mzPolyCldβ€˜π‘‰) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) β†’ ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f Β· 𝐺) ∈ 𝑃))
 
Theoremmzpval 41470 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝑉 ∈ V β†’ (mzPolyβ€˜π‘‰) = ∩ (mzPolyCldβ€˜π‘‰))
 
Theoremdmmzp 41471 mzPoly is defined for all index sets which are sets. This is used with elfvdm 6929 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
dom mzPoly = V
 
Theoremmzpincl 41472 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 β†’ (mzPolyβ€˜π‘‰) ∈ (mzPolyCldβ€˜π‘‰))
 
Theoremmzpconst 41473 Constant functions are polynomial. See also mzpconstmpt 41478. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝑉 ∈ V ∧ 𝐢 ∈ β„€) β†’ ((β„€ ↑m 𝑉) Γ— {𝐢}) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpf 41474 A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
(𝐹 ∈ (mzPolyβ€˜π‘‰) β†’ 𝐹:(β„€ ↑m 𝑉)βŸΆβ„€)
 
Theoremmzpproj 41475* A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝑉 ∈ V ∧ 𝑋 ∈ 𝑉) β†’ (𝑔 ∈ (β„€ ↑m 𝑉) ↦ (π‘”β€˜π‘‹)) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpadd 41476 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 41479. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ (mzPolyβ€˜π‘‰) ∧ 𝐡 ∈ (mzPolyβ€˜π‘‰)) β†’ (𝐴 ∘f + 𝐡) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpmul 41477 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 41480. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ (mzPolyβ€˜π‘‰) ∧ 𝐡 ∈ (mzPolyβ€˜π‘‰)) β†’ (𝐴 ∘f Β· 𝐡) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpconstmpt 41478* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 41479, mzpmulmpt 41480, mzpnegmpt 41482, mzpsubmpt 41481, mzpexpmpt 41483) 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 41475 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝑉 ∈ V ∧ 𝐢 ∈ β„€) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐢) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpaddmpt 41479* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 41476. (Contributed by Stefan O'Rear, 5-Oct-2014.)
(((π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐴) ∈ (mzPolyβ€˜π‘‰) ∧ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐡) ∈ (mzPolyβ€˜π‘‰)) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (𝐴 + 𝐡)) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpmulmpt 41480* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 41480. (Contributed by Stefan O'Rear, 5-Oct-2014.)
(((π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐴) ∈ (mzPolyβ€˜π‘‰) ∧ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐡) ∈ (mzPolyβ€˜π‘‰)) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (𝐴 Β· 𝐡)) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpsubmpt 41481* The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(((π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐴) ∈ (mzPolyβ€˜π‘‰) ∧ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐡) ∈ (mzPolyβ€˜π‘‰)) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (𝐴 βˆ’ 𝐡)) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpnegmpt 41482* Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
((π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐴) ∈ (mzPolyβ€˜π‘‰) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ -𝐴) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpexpmpt 41483* Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
(((π‘₯ ∈ (β„€ ↑m 𝑉) ↦ 𝐴) ∈ (mzPolyβ€˜π‘‰) ∧ 𝐷 ∈ β„•0) β†’ (π‘₯ ∈ (β„€ ↑m 𝑉) ↦ (𝐴↑𝐷)) ∈ (mzPolyβ€˜π‘‰))
 
Theoremmzpindd 41484* "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((πœ‘ ∧ 𝑓 ∈ β„€) β†’ πœ’)    &   ((πœ‘ ∧ 𝑓 ∈ 𝑉) β†’ πœƒ)    &   ((πœ‘ ∧ (𝑓:(β„€ ↑m 𝑉)βŸΆβ„€ ∧ 𝜏) ∧ (𝑔:(β„€ ↑m 𝑉)βŸΆβ„€ ∧ πœ‚)) β†’ 𝜁)    &   ((πœ‘ ∧ (𝑓:(β„€ ↑m 𝑉)βŸΆβ„€ ∧ 𝜏) ∧ (𝑔:(β„€ ↑m 𝑉)βŸΆβ„€ ∧ πœ‚)) β†’ 𝜎)    &   (π‘₯ = ((β„€ ↑m 𝑉) Γ— {𝑓}) β†’ (πœ“ ↔ πœ’))    &   (π‘₯ = (𝑔 ∈ (β„€ ↑m 𝑉) ↦ (π‘”β€˜π‘“)) β†’ (πœ“ ↔ πœƒ))    &   (π‘₯ = 𝑓 β†’ (πœ“ ↔ 𝜏))    &   (π‘₯ = 𝑔 β†’ (πœ“ ↔ πœ‚))    &   (π‘₯ = (𝑓 ∘f + 𝑔) β†’ (πœ“ ↔ 𝜁))    &   (π‘₯ = (𝑓 ∘f Β· 𝑔) β†’ (πœ“ ↔ 𝜎))    &   (π‘₯ = 𝐴 β†’ (πœ“ ↔ 𝜌))    β‡’   ((πœ‘ ∧ 𝐴 ∈ (mzPolyβ€˜π‘‰)) β†’ 𝜌)
 
Theoremmzpmfp 41485 Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)
(mzPolyβ€˜πΌ) = ran (𝐼 eval β„€ring)
 
Theoremmzpsubst 41486* 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.)
((π‘Š ∈ V ∧ 𝐹 ∈ (mzPolyβ€˜π‘‰) ∧ βˆ€π‘¦ ∈ 𝑉 𝐺 ∈ (mzPolyβ€˜π‘Š)) β†’ (π‘₯ ∈ (β„€ ↑m π‘Š) ↦ (πΉβ€˜(𝑦 ∈ 𝑉 ↦ (πΊβ€˜π‘₯)))) ∈ (mzPolyβ€˜π‘Š))
 
Theoremmzprename 41487* Simplified version of mzpsubst 41486 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
((π‘Š ∈ V ∧ 𝐹 ∈ (mzPolyβ€˜π‘‰) ∧ 𝑅:π‘‰βŸΆπ‘Š) β†’ (π‘₯ ∈ (β„€ ↑m π‘Š) ↦ (πΉβ€˜(π‘₯ ∘ 𝑅))) ∈ (mzPolyβ€˜π‘Š))
 
Theoremmzpresrename 41488* 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.)
((π‘Š ∈ V ∧ 𝑉 βŠ† π‘Š ∧ 𝐹 ∈ (mzPolyβ€˜π‘‰)) β†’ (π‘₯ ∈ (β„€ ↑m π‘Š) ↦ (πΉβ€˜(π‘₯ β†Ύ 𝑉))) ∈ (mzPolyβ€˜π‘Š))
 
Theoremmzpcompact2lem 41489* Lemma for mzpcompact2 41490. (Contributed by Stefan O'Rear, 9-Oct-2014.)
𝐡 ∈ V    β‡’   (𝐴 ∈ (mzPolyβ€˜π΅) β†’ βˆƒπ‘Ž ∈ Fin βˆƒπ‘ ∈ (mzPolyβ€˜π‘Ž)(π‘Ž βŠ† 𝐡 ∧ 𝐴 = (𝑐 ∈ (β„€ ↑m 𝐡) ↦ (π‘β€˜(𝑐 β†Ύ π‘Ž)))))
 
Theoremmzpcompact2 41490* 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 ∈ (mzPolyβ€˜π΅) β†’ βˆƒπ‘Ž ∈ Fin βˆƒπ‘ ∈ (mzPolyβ€˜π‘Ž)(π‘Ž βŠ† 𝐡 ∧ 𝐴 = (𝑐 ∈ (β„€ ↑m 𝐡) ↦ (π‘β€˜(𝑐 β†Ύ π‘Ž)))))
 
21.31.9  Miscellanea for Diophantine sets 1
 
Theoremcoeq0i 41491 coeq0 6255 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
((𝐴:𝐢⟢𝐷 ∧ 𝐡:𝐸⟢𝐹 ∧ (𝐢 ∩ 𝐹) = βˆ…) β†’ (𝐴 ∘ 𝐡) = βˆ…)
 
Theoremfzsplit1nn0 41492 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 ∈ β„•0 ∧ 𝐡 ∈ β„•0 ∧ 𝐴 ≀ 𝐡) β†’ (1...𝐡) = ((1...𝐴) βˆͺ ((𝐴 + 1)...𝐡)))
 
21.31.10  Diophantine sets 1: definitions
 
Syntaxcdioph 41493 Extend class notation to include the family of Diophantine sets.
class Dioph
 
Definitiondf-dioph 41494* 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 β„€ (via mzPoly) and β„•0 (to define the zero sets); the former could be avoided by considering coincidence sets of β„•0 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 16897 that implicitly restricting variables to β„•0 adds no expressive power over allowing them to range over β„€. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 41501. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Dioph = (𝑛 ∈ β„•0 ↦ ran (π‘˜ ∈ (β„€β‰₯β€˜π‘›), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0)}))
 
Theoremeldiophb 41495* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
(𝐷 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝐷 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
 
Theoremeldioph 41496* Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝑁 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑃 ∈ (mzPolyβ€˜(1...𝐾))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...𝐾))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
 
Theoremdiophrw 41497* 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 β†Ύ 𝑂) = ( I β†Ύ 𝑂)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m 𝑆)(π‘Ž = (𝑏 β†Ύ 𝑂) ∧ ((𝑑 ∈ (β„€ ↑m 𝑆) ↦ (π‘ƒβ€˜(𝑑 ∘ 𝑀)))β€˜π‘) = 0)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m 𝑇)(π‘Ž = (𝑐 β†Ύ 𝑂) ∧ (π‘ƒβ€˜π‘) = 0)})
 
Theoremeldioph2lem1 41498* Lemma for eldioph2 41500. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝑁 ∈ β„•0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) βŠ† 𝐴) β†’ βˆƒπ‘‘ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘’ ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
 
Theoremeldioph2lem2 41499* Lemma for eldioph2 41500. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
(((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ βˆƒπ‘(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
 
Theoremeldioph2 41500* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 41490. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
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