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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

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Last updated on 9-Feb-2025 at 6:50 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
28-Jan-2025dvdsrex 13079 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing → (∥r𝑅) ∈ V)
 
24-Jan-2025reldvdsrsrg 13073 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
(𝑅 ∈ SRing → Rel (∥r𝑅))
 
18-Jan-2025rerecapb 8776 A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ ℝ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
 
18-Jan-2025recapb 8604 A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1))
 
17-Jan-2025ressval3d 12500 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑𝐸 ∈ dom 𝑆)    &   (𝜑𝐴𝐵)       (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
17-Jan-2025strressid 12499 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑊)       (𝜑 → (𝑊s 𝐵) = 𝑊)
 
16-Jan-2025ressex 12494 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 12493 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
 
10-Jan-2025opprex 13057 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 13061. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (oppr𝑅)       (𝑅𝑉𝑂 ∈ V)
 
10-Jan-2025mgpex 12949 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 12964. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrp‘𝑅)       (𝑅𝑉𝑀 ∈ V)
 
5-Jan-2025imbibi 252 The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
(((𝜑𝜓) ↔ 𝜒) → (𝜑 → (𝜓𝜒)))
 
1-Jan-2025snss 3726 The singleton of an element of a class is a subset of the class (inference form of snssg 3725). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
𝐴 ∈ V       (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
 
1-Jan-2025snssg 3725 The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
(𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
 
1-Jan-2025snssb 3724 Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴𝐵))
 
9-Dec-2024nninfwlpoim 7169 Decidable equality for implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
(∀𝑥 ∈ ℕ𝑦 ∈ ℕ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
8-Dec-2024nninfwlpoimlemdc 7168 Lemma for nninfwlpoim 7169. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹𝑥) = ∅, ∅, 1o))    &   (𝜑 → ∀𝑥 ∈ ℕ𝑦 ∈ ℕ DECID 𝑥 = 𝑦)       (𝜑DECID𝑛 ∈ ω (𝐹𝑛) = 1o)
 
8-Dec-2024nninfwlpoimlemginf 7167 Lemma for nninfwlpoim 7169. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹𝑥) = ∅, ∅, 1o))       (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹𝑛) = 1o))
 
8-Dec-2024nninfwlpoimlemg 7166 Lemma for nninfwlpoim 7169. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹𝑥) = ∅, ∅, 1o))       (𝜑𝐺 ∈ ℕ)
 
7-Dec-2024nninfwlpor 7165 The Weak Limited Principle of Omniscience (WLPO) implies that equality for is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
(ω ∈ WOmni → ∀𝑥 ∈ ℕ𝑦 ∈ ℕ DECID 𝑥 = 𝑦)
 
7-Dec-2024nninfwlporlem 7164 Lemma for nninfwlpor 7165. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
(𝜑𝑋:ω⟶2o)    &   (𝜑𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))    &   (𝜑 → ω ∈ WOmni)       (𝜑DECID 𝑋 = 𝑌)
 
6-Dec-2024nninfwlporlemd 7163 Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
(𝜑𝑋:ω⟶2o)    &   (𝜑𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))       (𝜑 → (𝑋 = 𝑌𝐷 = (𝑖 ∈ ω ↦ 1o)))
 
3-Dec-2024nninfwlpo 7170 Decidability of equality for is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
(∀𝑥 ∈ ℕ𝑦 ∈ ℕ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni)
 
3-Dec-2024nninfdcinf 7162 The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
(𝜑 → ω ∈ WOmni)    &   (𝜑𝑁 ∈ ℕ)       (𝜑DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))
 
28-Nov-2024basmexd 12491 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐴𝐵)       (𝜑𝐺 ∈ V)
 
22-Nov-2024eliotaeu 5200 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5199 An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
 
18-Nov-2024basmex 12490 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)       (𝐴𝐵𝐺 ∈ V)
 
11-Nov-2024bj-con1st 14125 Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 
11-Nov-2024slotsdifdsndx 12622 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
 
11-Nov-2024tsetndxnstarvndx 12603 The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(TopSet‘ndx) ≠ (*𝑟‘ndx)
 
11-Nov-2024const 852 Contraposition when the antecedent is a negated stable proposition. See comment of condc 853. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)))
 
7-Nov-2024ressbasd 12496 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(𝜑𝑅 = (𝑊s 𝐴))    &   (𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) = (Base‘𝑅))
 
6-Nov-2024oppraddg 13060 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr𝑅)    &    + = (+g𝑅)       (𝑅𝑉+ = (+g𝑂))
 
6-Nov-2024opprbasg 13059 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝐵 = (Base‘𝑂))
 
6-Nov-2024opprsllem 13058 Lemma for opprbasg 13059 and oppraddg 13060. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (oppr𝑅)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (.r‘ndx)       (𝑅𝑉 → (𝐸𝑅) = (𝐸𝑂))
 
4-Nov-2024lgsfvalg 14039 Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐹𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))
 
1-Nov-2024qsqeqor 10603 The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))
 
31-Oct-2024dsndxnmulrndx 12619 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(dist‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnmulrndx 12602 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnbasendx 12600 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (Base‘ndx)
 
31-Oct-2024basendxlttsetndx 12599 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(Base‘ndx) < (TopSet‘ndx)
 
31-Oct-2024tsetndxnn 12598 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ∈ ℕ
 
29-Oct-2024dsndxntsetndx 12621 The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 12620 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024slotstnscsi 12604 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024scandxnmulrndx 12576 The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024fiubnn 10781 A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 ⊆ ℕ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℕ ∀𝑦𝐴 𝑦𝑥)
 
29-Oct-2024fiubz 10780 A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥)
 
29-Oct-2024fiubm 10779 Lemma for fiubz 10780 and fiubnn 10781. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℚ)    &   (𝜑𝐶𝐵)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → ∃𝑥𝐵𝑦𝐴 𝑦𝑥)
 
28-Oct-2024dsndxnbasendx 12617 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
(dist‘ndx) ≠ (Base‘ndx)
 
28-Oct-2024basendxltdsndx 12616 The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 12615 The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
27-Oct-2024bj-nnst 14117 Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 14364 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
¬ ¬ STAB 𝜑
 
27-Oct-2024bj-imnimnn 14112 If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 14111 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑𝜓)    &   𝜑𝜓)        ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 12010 Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓𝑥𝑦)))
 
23-Oct-2024nnwodc 12007 Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.)
((𝐴 ⊆ ℕ ∧ ∃𝑤 𝑤𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗𝐴) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
 
22-Oct-2024uzwodc 12008 Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.)
((𝑆 ⊆ (ℤ𝑀) ∧ ∃𝑥 𝑥𝑆 ∧ ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝑆) → ∃𝑗𝑆𝑘𝑆 𝑗𝑘)
 
21-Oct-2024nnnotnotr 14364 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 850, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
¬ ¬ (¬ ¬ 𝜑𝜑)
 
21-Oct-2024scandxnbasendx 12574 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalar‘ndx) ≠ (Base‘ndx)
 
20-Oct-2024isprm5lem 12111 Lemma for isprm5 12112. The interesting direction (showing that one only needs to check prime divisors up to the square root of 𝑃). (Contributed by Jim Kingdon, 20-Oct-2024.)
(𝜑𝑃 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧𝑃))    &   (𝜑𝑋 ∈ (2...(𝑃 − 1)))       (𝜑 → ¬ 𝑋𝑃)
 
19-Oct-2024resseqnbasd 12501 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (𝑊s 𝐴)    &   𝐶 = (𝐸𝑊)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (Base‘ndx)    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝑉)       (𝜑𝐶 = (𝐸𝑅))
 
18-Oct-2024dsndxnplusgndx 12618 The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024tsetndxnplusgndx 12601 The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(TopSet‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024scandxnplusgndx 12575 The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(Scalar‘ndx) ≠ (+g‘ndx)
 
17-Oct-2024elnndc 9588 Membership of an integer in is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 11222 Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) = if(𝐴𝐵, 𝐴, 𝐵))
 
14-Oct-2024mingeb 11221 Equivalence of and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴))
 
13-Oct-2024pcxnn0cl 12280 Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt 𝑁) ∈ ℕ0*)
 
13-Oct-2024xnn0letri 9777 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴𝐵𝐵𝐴))
 
13-Oct-2024xnn0dcle 9776 Decidability of for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → DECID 𝐴𝐵)
 
9-Oct-2024nn0leexp2 10662 Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀𝑁 ↔ (𝐴𝑀) ≤ (𝐴𝑁)))
 
8-Oct-2024pclemdc 12258 Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}       ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∀𝑥 ∈ ℤ DECID 𝑥𝐴)
 
8-Oct-2024elnn0dc 9587 Membership of an integer in 0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 12257 Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}       ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥)
 
7-Oct-2024pclem0 12256 Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑁}       ((𝑃 ∈ (ℤ‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴)
 
7-Oct-2024nn0ltexp2 10661 Special case of ltexp2 13993 which we use here because we haven't yet defined df-rpcxp 13913 which is used in the current proof of ltexp2 13993. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴𝑀) < (𝐴𝑁)))
 
6-Oct-2024suprzcl2dc 11926 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7910.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
(𝜑𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑 → ∃𝑥 𝑥𝐴)       (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
 
5-Oct-2024zsupssdc 11925 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7910.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(𝜑𝐴 ⊆ ℤ)    &   (𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥)       (𝜑 → ∃𝑥𝐴 (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦𝐵 (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
5-Oct-2024suprzubdc 11923 The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(𝜑𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))
 
1-Oct-2024infex2g 7026 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 12425 An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ)
 
30-Sep-2024prmdc 12100 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 6973 Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 DECID 𝜑) → DECID𝑥𝐴 𝜑)
 
29-Sep-2024ssnnct 12418 A decidable subset of is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
29-Sep-2024ssnnctlemct 12417 Lemma for ssnnct 12418. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)       ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
28-Sep-2024nninfdcex 11924 A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∃𝑦 𝑦𝐴)       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
 
27-Sep-2024infregelbex 9574 Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 𝐵𝑧))
 
26-Sep-2024nninfdclemp1 12421 Lemma for nninfdc 12424. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑𝑈 ∈ ℕ)       (𝜑 → (𝐹𝑈) < (𝐹‘(𝑈 + 1)))
 
26-Sep-2024nnminle 12006 The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12005. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 12419 Lemma for nninfdc 12424. (Contributed by Jim Kingdon, 25-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑃(𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴)
 
24-Sep-2024nninfdclemlt 12422 Lemma for nninfdc 12424. The function from nninfdclemf 12420 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑𝑈 ∈ ℕ)    &   (𝜑𝑉 ∈ ℕ)    &   (𝜑𝑈 < 𝑉)       (𝜑 → (𝐹𝑈) < (𝐹𝑉))
 
23-Sep-2024nninfdc 12424 An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → ω ≼ 𝐴)
 
23-Sep-2024nninfdclemf1 12423 Lemma for nninfdc 12424. The function from nninfdclemf 12420 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))       (𝜑𝐹:ℕ–1-1𝐴)
 
23-Sep-2024nninfdclemf 12420 Lemma for nninfdc 12424. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))       (𝜑𝐹:ℕ⟶𝐴)
 
23-Sep-2024nnmindc 12005 An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∃𝑦 𝑦𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴)
 
19-Sep-2024ssomct 12416 A decidable subset of ω is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
14-Sep-2024nnpredlt 4619 The predecessor (see nnpredcl 4618) of a nonzero natural number is less than (see df-iord 4362) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
 
13-Sep-2024nninfisollemeq 7123 Lemma for nninfisol 7124. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑𝑋 ∈ ℕ)    &   (𝜑 → (𝑋𝑁) = ∅)    &   (𝜑𝑁 ∈ ω)    &   (𝜑𝑁 ≠ ∅)    &   (𝜑 → (𝑋 𝑁) = 1o)       (𝜑DECID (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollemne 7122 Lemma for nninfisol 7124. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑𝑋 ∈ ℕ)    &   (𝜑 → (𝑋𝑁) = ∅)    &   (𝜑𝑁 ∈ ω)    &   (𝜑𝑁 ≠ ∅)    &   (𝜑 → (𝑋 𝑁) = ∅)       (𝜑DECID (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollem0 7121 Lemma for nninfisol 7124. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑𝑋 ∈ ℕ)    &   (𝜑 → (𝑋𝑁) = ∅)    &   (𝜑𝑁 ∈ ω)    &   (𝜑𝑁 = ∅)       (𝜑DECID (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)) = 𝑋)

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