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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.

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Last updated on 21-Dec-2024 at 6:47 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
9-Dec-2024nninfwlpoim 7154 Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
(∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
8-Dec-2024nninfwlpoimlemdc 7153 Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    &   (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)    ⇒   (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)
 
8-Dec-2024nninfwlpoimlemginf 7152 Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o))
 
8-Dec-2024nninfwlpoimlemg 7151 Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   (𝜑 → 𝐺 ∈ ℕ∞)
 
7-Dec-2024nninfwlpor 7150 The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
(ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)
 
7-Dec-2024nninfwlporlem 7149 Lemma for nninfwlpor 7150. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
(𝜑 → 𝑋:ω⟶2o)    &   (𝜑 → 𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    &   (𝜑 → ω ∈ WOmni)    ⇒   (𝜑 → DECID 𝑋 = 𝑌)
 
6-Dec-2024nninfwlporlemd 7148 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.)
(𝜑 → 𝑋:ω⟶2o)    &   (𝜑 → 𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    ⇒   (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
 
3-Dec-2024nninfwlpo 7155 Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
(∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni)
 
3-Dec-2024nninfdcinf 7147 The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
(𝜑 → ω ∈ WOmni)    &   (𝜑 → 𝑁 ∈ ℕ∞)    ⇒   (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))
 
28-Nov-2024basmexd 12475 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(𝜑 → 𝐵 = (Base‘𝐺))    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → 𝐺 ∈ V)
 
22-Nov-2024eliotaeu 5187 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5186 An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
 
18-Nov-2024basmex 12474 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝐵 → 𝐺 ∈ V)
 
11-Nov-2024bj-con1st 13786 Contraposition when the antecedent is a negated stable proposition. See con1dc 851. (Contributed by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
 
11-Nov-2024const 847 Contraposition when the antecedent is a negated stable proposition. See comment of condc 848. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
4-Nov-2024lgsfvalg 13700 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) → (𝐹‘𝑀) = 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 10586 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 = 𝐵 ∨ 𝐴 = -𝐵)))
 
29-Oct-2024fiubnn 10765 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 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
29-Oct-2024fiubz 10764 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 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
29-Oct-2024fiubm 10763 Lemma for fiubz 10764 and fiubnn 10765. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐵 ⊆ ℚ)    &   (𝜑 → 𝐶 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
27-Oct-2024bj-nnst 13778 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 14025 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 105, ax-ia2 106, ax-ia3 107 are via sylibr 133, 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 13773 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 13772 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒    ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 11994 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 → (𝜑 ↔ 𝜓))    ⇒   ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
 
23-Oct-2024nnwodc 11991 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 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
 
22-Oct-2024uzwodc 11992 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 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑥 𝑥 ∈ 𝑆 ∧ ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝑆) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
 
21-Oct-2024nnnotnotr 14025 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 845, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
¬ ¬ (¬ ¬ 𝜑 → 𝜑)
 
20-Oct-2024isprm5lem 12095 Lemma for isprm5 12096. 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.)
(𝜑 → 𝑃 ∈ (ℤ≥‘2))    &   (𝜑 → ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))    &   (𝜑 → 𝑋 ∈ (2...(𝑃 − 1)))    ⇒   (𝜑 → ¬ 𝑋 ∥ 𝑃)
 
17-Oct-2024elnndc 9571 Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 11206 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(𝐴 ≤ 𝐵, 𝐴, 𝐵))
 
14-Oct-2024mingeb 11205 Equivalence of ≤ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴))
 
13-Oct-2024pcxnn0cl 12264 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 9760 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
13-Oct-2024xnn0dcle 9759 Decidability of ≤ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → DECID 𝐴 ≤ 𝐵)
 
9-Oct-2024nn0leexp2 10645 Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))
 
8-Oct-2024pclemdc 12242 Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
 
8-Oct-2024elnn0dc 9570 Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 12241 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 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
7-Oct-2024pclem0 12240 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 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴)
 
7-Oct-2024nn0ltexp2 10644 Special case of ltexp2 13654 which we use here because we haven't yet defined df-rpcxp 13574 which is used in the current proof of ltexp2 13654. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 11910 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7895.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    ⇒   (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
 
5-Oct-2024zsupssdc 11909 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7895.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
5-Oct-2024suprzubdc 11907 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.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
 
1-Oct-2024infex2g 7011 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 12409 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 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ)
 
30-Sep-2024prmdc 12084 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 6958 Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∀𝑥 ∈ 𝐴 𝜑)
 
29-Sep-2024ssnnct 12402 A decidable subset of ℕ is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
29-Sep-2024ssnnctlemct 12401 Lemma for ssnnct 12402. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)    ⇒   ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
28-Sep-2024nninfdcex 11908 A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑦 𝑦 ∈ 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
 
27-Sep-2024infregelbex 9557 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 (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   (𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))
 
26-Sep-2024nninfdclemp1 12405 Lemma for nninfdc 12408. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑 → 𝑈 ∈ ℕ)    ⇒   (𝜑 → (𝐹‘𝑈) < (𝐹‘(𝑈 + 1)))
 
26-Sep-2024nnminle 11990 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 11989. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 12403 Lemma for nninfdc 12408. (Contributed by Jim Kingdon, 25-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → 𝑃 ∈ 𝐴)    &   (𝜑 → 𝑄 ∈ 𝐴)    ⇒   (𝜑 → (𝑃(𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴)
 
24-Sep-2024nninfdclemlt 12406 Lemma for nninfdc 12408. The function from nninfdclemf 12404 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑 → 𝑈 ∈ ℕ)    &   (𝜑 → 𝑉 ∈ ℕ)    &   (𝜑 → 𝑈 < 𝑉)    ⇒   (𝜑 → (𝐹‘𝑈) < (𝐹‘𝑉))
 
23-Sep-2024nninfdc 12408 An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ω ≼ 𝐴)
 
23-Sep-2024nninfdclemf1 12407 Lemma for nninfdc 12408. The function from nninfdclemf 12404 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    ⇒   (𝜑 → 𝐹:ℕ–1-1→𝐴)
 
23-Sep-2024nninfdclemf 12404 Lemma for nninfdc 12408. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    ⇒   (𝜑 → 𝐹:ℕ⟶𝐴)
 
23-Sep-2024nnmindc 11989 An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴)
 
19-Sep-2024ssomct 12400 A decidable subset of ω is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
14-Sep-2024nnpredlt 4608 The predecessor (see nnpredcl 4607) of a nonzero natural number is less than (see df-iord 4351) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7108 Lemma for nninfisol 7109. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 ≠ ∅)    &   (𝜑 → (𝑋‘∪ 𝑁) = 1o)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollemne 7107 Lemma for nninfisol 7109. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 ≠ ∅)    &   (𝜑 → (𝑋‘∪ 𝑁) = ∅)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollem0 7106 Lemma for nninfisol 7109. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 = ∅)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
12-Sep-2024nninfisol 7109 Finite elements of ℕ∞ are isolated. That is, given a natural number and any element of ℕ∞, it is decidable whether the natural number (when converted to an element of ℕ∞) is equal to the given element of ℕ∞. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence 𝑋 to decide whether it is equal to 𝑁 (in fact, you only need to look at two elements and 𝑁 tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
7-Sep-2024eulerthlemfi 12182 Lemma for eulerth 12187. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
7-Sep-2024modqexp 10602 Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝐵 ∈ ℤ)    &   (𝜑 → 𝐶 ∈ ℕ0)    &   (𝜑 → 𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))    ⇒   (𝜑 → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
 
5-Sep-2024eulerthlemh 12185 Lemma for eulerth 12187. A permutation of (1...(ϕ‘𝑁)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    &   𝐻 = (◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))    ⇒   (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
 
2-Sep-2024eulerthlemth 12186 Lemma for eulerth 12187. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))
 
2-Sep-2024eulerthlema 12184 Lemma for eulerth 12187. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁))
 
2-Sep-2024eulerthlemrprm 12183 Lemma for eulerth 12187. 𝑁 and ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
 
30-Aug-2024fprodap0f 11599 A finite product of terms apart from zero is apart from zero. A version of fprodap0 11584 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
Ⅎ𝑘𝜑    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 # 0)
 
28-Aug-2024fprodrec 11592 The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
 
26-Aug-2024exmidontri2or 7220 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
26-Aug-2024exmidontri 7216 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
26-Aug-2024ontri2orexmidim 4556 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4555. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
 
26-Aug-2024ontriexmidim 4506 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4505. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
 
25-Aug-2024onntri2or 7223 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
25-Aug-2024onntri3or 7222 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
25-Aug-2024csbcow 3060 Composition law for chained substitutions into a class. Version of csbco 3059 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.)
⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
 
25-Aug-2024cbvreuvw 2702 Version of cbvreuv 2698 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvrexvw 2701 Version of cbvrexv 2697 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvralvw 2700 Version of cbvralv 2696 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvabw 2293 Version of cbvab 2294 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
 
25-Aug-2024nfsbv 1940 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 1939 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
25-Aug-2024cbvexvw 1913 Change bound variable. See cbvexv 1911 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1441. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
25-Aug-2024cbvalvw 1912 Change bound variable. See cbvalv 1910 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1441. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
25-Aug-2024nfal 1569 If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1503. (Revised by Gino Giotto, 25-Aug-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦𝜑
 
24-Aug-2024gcdcomd 11929 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
21-Aug-2024dvds2addd 11791 Deduction form of dvds2add 11787. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 + 𝑁))
 
17-Aug-2024fprodcl2lem 11568 Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
(𝜑 → 𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
 
16-Aug-2024if0ab 13840 Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion if(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3574, (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 13841 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
 
16-Aug-2024fprodunsn 11567 Multiply in an additional term in a finite product. See also fprodsplitsn 11596 which is the same but with a Ⅎ𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.)
Ⅎ𝑘𝐷    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    &   (𝑘 = 𝐵 → 𝐶 = 𝐷)    ⇒   (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
 
15-Aug-2024bj-charfundcALT 13844 Alternate proof of bj-charfundc 13843. It was expected to be much shorter since it uses bj-charfun 13842 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
15-Aug-2024bj-charfun 13842 Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    ⇒   (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
15-Aug-2024fmelpw1o 13841 With a formula 𝜑 one can associate an element of 𝒫 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than ⊤ and ⊥, by nndc 846, which translate to 1o and ∅ respectively by iftrue 3531 and iffalse 3534, giving pwtrufal 14030).

As proved in if0ab 13840, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.)

if(𝜑, 1o, ∅) ∈ 𝒫 1o
 
15-Aug-2024cnstab 8564 Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 826. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵)
 
15-Aug-2024subap0d 8563 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 𝐵)    ⇒   (𝜑 → (𝐴 − 𝐵) # 0)
 
15-Aug-2024ifexd 4469 Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
 
15-Aug-2024ifelpwun 4468 Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)
 
15-Aug-2024ifelpwund 4467 Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
 
15-Aug-2024ifelpwung 4466 Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
 
15-Aug-2024ifidss 3541 A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐴) ⊆ 𝐴
 
15-Aug-2024ifssun 3540 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
 
12-Aug-2024exmidontriimlem2 7199 Lemma for exmidontriim 7202. (Contributed by Jim Kingdon, 12-Aug-2024.)
(𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑦 ∈ 𝐵 𝑦 ∈ 𝐴))
 
12-Aug-2024exmidontriimlem1 7198 Lemma for exmidontriim 7202. A variation of r19.30dc 2617. (Contributed by Jim Kingdon, 12-Aug-2024.)
((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒))
 
11-Aug-2024nndc 846 Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula.

This should not trick the reader into thinking that ¬ ¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 845 over 𝜑 would give "∀𝜑¬ ¬ DECID 𝜑", but EXMID is "∀𝜑DECID 𝜑", so proving ¬ ¬ EXMID would amount to proving "¬ ¬ ∀𝜑DECID 𝜑", which is not implied by the above theorem. Indeed, the converse of nnal 1642 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of 𝒫 1o, like we do in our definition of EXMID (df-exmid 4181): then, we can prove ∀𝑥 ∈ 𝒫 1o¬ ¬ DECID 𝑥 = 1o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o because the converse of nnral 2460 does not hold.

Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 688). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)

¬ ¬ DECID 𝜑
 
10-Aug-2024exmidontriim 7202 Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
(EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
10-Aug-2024exmidontriimlem4 7201 Lemma for exmidontriim 7202. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
 
10-Aug-2024exmidontriimlem3 7200 Lemma for exmidontriim 7202. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    &   (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
 
10-Aug-2024nnnninf2 7103 Canonical embedding of suc ω into ℕ∞. (Contributed by BJ, 10-Aug-2024.)
(𝑁 ∈ suc ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
 
10-Aug-2024infnninf 7100 The point at infinity in ℕ∞ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1o}), as fconstmpt 4658 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
(𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
 
9-Aug-2024ss1o0el1o 6890 Reformulation of ss1o0el1 4183 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 6889 Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
 
9-Aug-2024ss1o0el1 4183 A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
(𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))
 
8-Aug-2024pw1dc1 6891 If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
 
7-Aug-2024pw1fin 6888 Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
(EXMID ↔ 𝒫 1o ∈ Fin)
 
7-Aug-2024elomssom 4589 A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4590. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ ω → 𝐴 ⊆ ω)
 
6-Aug-2024bj-charfunbi 13846 In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function.

This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

(𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
 
6-Aug-2024bj-charfunr 13845 If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋.

The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

(𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
 
6-Aug-2024bj-charfundc 13843 Properties of the characteristic function on the class 𝑋 of the class 𝐴, provided membership in 𝐴 is decidable in 𝑋. (Contributed by BJ, 6-Aug-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
6-Aug-2024prodssdc 11552 Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))) ⇝ 𝑦))    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)    &   (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
 
5-Aug-2024fnmptd 13839 The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → 𝐹 Fn 𝐴)
 
5-Aug-2024funmptd 13838 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5235, then prove funmptd 13838 from it, and then prove funmpt 5236 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

(𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   (𝜑 → Fun 𝐹)
 
5-Aug-2024bj-dcfal 13790 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊥
 
5-Aug-2024bj-dctru 13788 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊤
 
5-Aug-2024bj-stfal 13777 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊥
 
5-Aug-2024bj-sttru 13775 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊤
 
5-Aug-2024prod1dc 11549 Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)
(((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘 ∈ 𝐴 1 = 1)
 
5-Aug-20242ssom 6503 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
2-Aug-2024onntri52 7221 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
2-Aug-2024onntri24 7219 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
2-Aug-2024onntri45 7218 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)
 
2-Aug-2024onntri51 7217 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
2-Aug-2024onntri13 7215 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
2-Aug-2024onntri35 7214 Double negated ordinal trichotomy.

There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7215), (3) implies (5) (onntri35 7214), (5) implies (1) (onntri51 7217), (2) implies (4) (onntri24 7219), (4) implies (5) (onntri45 7218), and (5) implies (2) (onntri52 7221).

Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7216 and exmidontri2or 7220, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7222 and (4) by onntri2or 7223.

(Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)
 
1-Aug-2024nnral 2460 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1642. (Contributed by Jim Kingdon, 1-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
 
31-Jul-20243nsssucpw1 7213 Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
(¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)
 
31-Jul-2024sucpw1nss3 7212 Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
(¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
 
30-Jul-20243nelsucpw1 7211 Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
¬ 3o ∈ suc 𝒫 1o
 
30-Jul-2024sucpw1nel3 7210 The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
¬ suc 𝒫 1o ∈ 3o
 
30-Jul-2024sucpw1ne3 7209 Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
(¬ EXMID → suc 𝒫 1o ≠ 3o)
 
30-Jul-2024pw1nel3 7208 Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
(¬ EXMID → ¬ 𝒫 1o ∈ 3o)
 
30-Jul-2024pw1ne3 7207 The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 3o
 
30-Jul-2024pw1ne1 7206 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o
 
30-Jul-2024pw1ne0 7205 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅
 
29-Jul-2024grpcld 12721 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
29-Jul-2024pw1on 7203 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
28-Jul-2024exmidpweq 6887 Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
(EXMID ↔ 𝒫 1o = 2o)
 
27-Jul-2024dcapnconstALT 14093 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 14092 by means of dceqnconst 14091. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
27-Jul-2024reap0 14090 Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℝ DECID 𝑧 # 0)
 
26-Jul-2024nconstwlpolemgt0 14095 Lemma for nconstwlpo 14097. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    &   (𝜑 → ∃𝑥 ∈ ℕ (𝐺‘𝑥) = 1)    ⇒   (𝜑 → 0 < 𝐴)
 
26-Jul-2024nconstwlpolem0 14094 Lemma for nconstwlpo 14097. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    &   (𝜑 → ∀𝑥 ∈ ℕ (𝐺‘𝑥) = 0)    ⇒   (𝜑 → 𝐴 = 0)
 
24-Jul-2024tridceq 14088 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14075 and redcwlpo 14087). Thus, this is an analytic analogue to lpowlpo 7144. (Contributed by Jim Kingdon, 24-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
 
24-Jul-2024iswomni0 14083 Weak omniscience stated in terms of equality with 0. Like iswomninn 14082 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
 
24-Jul-2024lpowlpo 7144 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7143. There is an analogue in terms of analytic omniscience principles at tridceq 14088. (Contributed by Jim Kingdon, 24-Jul-2024.)
(ω ∈ Omni → ω ∈ WOmni)
 
23-Jul-2024nconstwlpolem 14096 Lemma for nconstwlpo 14097. (Contributed by Jim Kingdon, 23-Jul-2024.)
(𝜑 → 𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0)    &   (𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    ⇒   (𝜑 → (∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0 ∨ ¬ ∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0))
 
23-Jul-2024dceqnconst 14091 Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 14087 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
(∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
23-Jul-2024redc0 14089 Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
 
23-Jul-2024canth 5807 No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1493 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
𝐴 ∈ V    ⇒    ¬ 𝐹:𝐴–onto→𝒫 𝐴
 
22-Jul-2024nconstwlpo 14097 Existence of a certain non-constant function from reals to integers implies ω ∈ WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
(𝜑 → 𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0)    ⇒   (𝜑 → ω ∈ WOmni)
 
15-Jul-2024fprodseq 11546 The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
(𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 1)))‘𝑀))
 
14-Jul-2024rexbid2 2475 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
 
14-Jul-2024ralbid2 2474 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
 
12-Jul-20242irrexpqap 13690 There exist real numbers 𝑎 and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (𝑎↑𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers (√‘2) and (2 logb 9), see sqrt2irrap 12134, 2logb9irrap 13689 and sqrt2cxp2logb9e3 13687. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ)
 
12-Jul-20242logb9irrap 13689 Example for logbgcd1irrap 13682. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
(𝑄 ∈ ℚ → (2 logb 9) # 𝑄)
 
11-Jul-2024logbgcd1irraplemexp 13680 Lemma for logbgcd1irrap 13682. Apartness of 𝑋↑𝑁 and 𝐵↑𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.)
(𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀))
 
11-Jul-2024reapef 13493 Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (exp‘𝐴) # (exp‘𝐵)))
 
10-Jul-2024apcxp2 13652 Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
(((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 # 𝐶 ↔ (𝐴↑𝑐𝐵) # (𝐴↑𝑐𝐶)))
 
9-Jul-2024logbgcd1irraplemap 13681 Lemma for logbgcd1irrap 13682. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
(𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁))
 
9-Jul-2024apexp1 10652 Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # (𝐵↑𝑁) → 𝐴 # 𝐵))
 
5-Jul-2024logrpap0 13592 The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) → (log‘𝐴) # 0)
 
3-Jul-2024rplogbval 13657 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)
((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))
 
3-Jul-2024logrpap0d 13593 Deduction form of logrpap0 13592. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → (log‘𝐴) # 0)
 
3-Jul-2024logrpap0b 13591 The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝐴 ∈ ℝ+ → (𝐴 # 1 ↔ (log‘𝐴) # 0))
 
28-Jun-20242o01f 14029 Mapping zero and one between ω and ℕ0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐺 ↾ 2o):2o⟶{0, 1}
 
28-Jun-2024012of 14028 Mapping zero and one between ℕ0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o
 
27-Jun-2024iooreen 14067 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
27-Jun-2024iooref1o 14066 A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
𝐹 = (𝑥 ∈ ℝ ↦ (1 / (1 + (exp‘𝑥))))    ⇒   𝐹:ℝ–1-1-onto→(0(,)1)
 
25-Jun-2024neapmkvlem 14098 Lemma for neapmkv 14099. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   ((𝜑 ∧ 𝐴 ≠ 1) → 𝐴 # 1)    ⇒   (𝜑 → (¬ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
 
25-Jun-2024ismkvnn 14085 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0)))
 
25-Jun-2024ismkvnnlem 14084 Lemma for ismkvnn 14085. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0)))
 
25-Jun-2024enmkvlem 7137 Lemma for enmkv 7138. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
 
24-Jun-2024neapmkv 14099 If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) → ω ∈ Markov)
 
24-Jun-2024dcapnconst 14092 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 14075 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 14091 and in fact this theorem can be proved using dceqnconst 14091 as shown at dcapnconstALT 14093. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
24-Jun-2024enmkv 7138 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either ω ∈ Markov or ℕ0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6409 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov))
 
21-Jun-2024redcwlpolemeq1 14086 Lemma for redcwlpo 14087. A biconditionalized version of trilpolemeq1 14072. (Contributed by Jim Kingdon, 21-Jun-2024.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    ⇒   (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1))
 
20-Jun-2024redcwlpo 14087 Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 14086). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10203 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
20-Jun-2024iswomninn 14082 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7142 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
 
20-Jun-2024iswomninnlem 14081 Lemma for iswomnimap 7142. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
 
20-Jun-2024enwomni 7146 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either ω ∈ WOmni or ℕ0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6409 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni ↔ 𝐵 ∈ WOmni))
 
20-Jun-2024enwomnilem 7145 Lemma for enwomni 7146. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 13653 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 0 < (ℜ‘𝐵))    &   (𝜑 → 𝑀 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) ≤ 𝑀)    ⇒   (𝜑 → (abs‘(𝐴↑𝑐𝐵)) ≤ ((𝑀↑𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
 
16-Jun-2024rpcxpsqrt 13636 The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
(𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) = (√‘𝐴))
 
13-Jun-2024rpcxpadd 13620 Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐(𝐵 + 𝐶)) = ((𝐴↑𝑐𝐵) · (𝐴↑𝑐𝐶)))
 
12-Jun-2024cxpap0 13619 Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) # 0)
 
12-Jun-2024rpcncxpcl 13617 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ)
 
12-Jun-2024rpcxp0 13613 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
(𝐴 ∈ ℝ+ → (𝐴↑𝑐0) = 1)
 
12-Jun-2024cxpexpnn 13611 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵))
 
12-Jun-2024cxpexprp 13610 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵))
 
12-Jun-2024rpcxpef 13609 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
 
12-Jun-2024df-rpcxp 13574 Define the power function on complex numbers. Because df-relog 13573 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.)
↑𝑐 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
 
10-Jun-2024trirec0xor 14077 Version of trirec0 14076 with exclusive-or.

The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))
 
10-Jun-2024trirec0 14076 Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy.

This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 14075). (Contributed by Jim Kingdon, 10-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
 
9-Jun-2024omniwomnimkv 7143 A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω says that LPO ↔ WLPO ∧ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))
 
9-Jun-2024iswomnimap 7142 The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
 
9-Jun-2024iswomni 7141 The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
 
9-Jun-2024df-womni 7140 A weakly omniscient set is one where we can decide whether a predicate (here represented by a function 𝑓) holds (is equal to 1o) for all elements or not. Generalization of definition 2.4 of [Pierik], p. 9.

In particular, ω ∈ WOmni is known as the Weak Limited Principle of Omniscience (WLPO).

The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.)

WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)}
 
1-Jun-2024grpmndd 12720 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
29-May-2024pw1nct 14036 A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
(∀𝑟(𝑟 ⊆ (𝒫 1o × ω) → (∀𝑝 ∈ 𝒫 1o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o))
 
28-May-2024sssneq 14035 Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
(𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
 
26-May-2024elpwi2 4144 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
𝐵 ∈ 𝑉    &   𝐴 ⊆ 𝐵    ⇒   𝐴 ∈ 𝒫 𝐵
 
24-May-2024dvmptcjx 13480 Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   (𝜑 → 𝑋 ⊆ ℝ)    ⇒   (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵)))
 
23-May-2024cbvralfw 2687 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2689 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1500 and ax-bndl 1502 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
22-May-2024efltlemlt 13489 Lemma for eflt 13490. The converse of efltim 11661 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (exp‘𝐴) < (exp‘𝐵))    &   (𝜑 → 𝐷 ∈ ℝ+)    &   (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐷 → (abs‘((exp‘𝐴) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))    ⇒   (𝜑 → 𝐴 < 𝐵)
 
21-May-2024eflt 13490 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵)))
 
19-May-2024apdifflemr 14079 Lemma for apdiff 14080. (Contributed by Jim Kingdon, 19-May-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑆 ∈ ℚ)    &   (𝜑 → (abs‘(𝐴 − -1)) # (abs‘(𝐴 − 1)))    &   ((𝜑 ∧ 𝑆 ≠ 0) → (abs‘(𝐴 − 0)) # (abs‘(𝐴 − (2 · 𝑆))))    ⇒   (𝜑 → 𝐴 # 𝑆)
 
18-May-2024apdifflemf 14078 Lemma for apdiff 14080. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑄 ∈ ℚ)    &   (𝜑 → 𝑅 ∈ ℚ)    &   (𝜑 → 𝑄 < 𝑅)    &   (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)    ⇒   (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
 
17-May-2024apdiff 14080 The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
(𝐴 ∈ ℝ → (∀𝑞 ∈ ℚ 𝐴 # 𝑞 ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) # (abs‘(𝐴 − 𝑟)))))
 
15-May-2024reeff1oleme 13487 Lemma for reeff1o 13488. (Contributed by Jim Kingdon, 15-May-2024.)
(𝑈 ∈ (0(,)e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
14-May-2024df-relog 13573 Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.)
log = ◡(exp ↾ ℝ)
 
12-May-2024dvdstrd 11792 The divides relation is transitive, a deduction version of dvdstr 11790. (Contributed by metakunt, 12-May-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝑀 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ 𝑁)
 
7-May-2024ioocosf1o 13569 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.)
(cos ↾ (0(,)π)):(0(,)π)–1-1-onto→(-1(,)1)
 
7-May-2024cos0pilt1 13567 Cosine is between minus one and one on the open interval between zero and π. (Contributed by Jim Kingdon, 7-May-2024.)
(𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1))
 
6-May-2024cos11 13568 Cosine is one-to-one over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵)))
 
5-May-2024omiunct 12399 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12395 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑 → CCHOICE)    &   ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o))
 
5-May-2024ctiunctal 12396 Variation of ctiunct 12395 which allows 𝑥 to be present in 𝜑. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   (𝜑 → ∀𝑥 ∈ 𝐴 𝐺:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
 
3-May-2024cc4n 7233 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7232, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
3-May-2024cc4f 7231 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   Ⅎ𝑛𝐴    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
1-May-2024cc4 7232 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
29-Apr-2024cc3 7230 Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)    &   (𝜑 → 𝑁 ≈ ω)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
 
27-Apr-2024cc2 7229 Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    ⇒   (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
 
27-Apr-2024cc2lem 7228 Lemma for cc2 7229. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    &   𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))    &   𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛))))    ⇒   (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
 
27-Apr-2024cc1 7227 Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))
 
19-Apr-2024omctfn 12398 Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
(𝜑 → CCHOICE)    &   ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃𝑓(𝑓 Fn ω ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)))
 
13-Apr-2024prodmodclem2 11540 Lemma for prodmodc 11541. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    ⇒   ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
 
11-Apr-2024prodmodclem2a 11539 Lemma for prodmodc 11541. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))    ⇒   (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))
 
11-Apr-2024prodmodclem3 11538 Lemma for prodmodc 11541. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)    ⇒   (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
 
10-Apr-2024jcnd 647 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → ¬ (𝜓 → 𝜒))
 
4-Apr-2024prodrbdclem 11534 Lemma for prodrbdc 11537. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( · , 𝐹))
 
24-Mar-2024prodfdivap 11510 The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) # 0)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))
 
24-Mar-2024prodfrecap 11509 The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    ⇒   (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
 
23-Mar-2024prodfap0 11508 The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)    ⇒   (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)
 
22-Mar-2024prod3fmul 11504 The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , 𝐺)‘𝑁)))
 
21-Mar-2024df-proddc 11514 Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sumdc 11317 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.)
∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
 
19-Mar-2024cos02pilt1 13566 Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)
 
19-Mar-2024cosq34lt1 13565 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (π[,)(2 · π)) → (cos‘𝐴) < 1)
 
14-Mar-2024coseq0q4123 13549 Location of the zeroes of cosine in (-(π / 2)(,)(3 · (π / 2))). (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2)))
 
14-Mar-2024cosq23lt0 13548 The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)
 
9-Mar-2024pilem3 13498 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)
 
9-Mar-2024exmidonfin 7171 If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6850 and nnon 4594. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(ω = (On ∩ Fin) → EXMID)
 
9-Mar-2024exmidonfinlem 7170 Lemma for exmidonfin 7171. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   (ω = (On ∩ Fin) → DECID 𝜑)
 
8-Mar-2024sin0pilem2 13497 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
∃𝑞 ∈ (2(,)4)((sin‘𝑞) = 0 ∧ ∀𝑥 ∈ (0(,)𝑞)0 < (sin‘𝑥))
 
8-Mar-2024sin0pilem1 13496 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
∃𝑝 ∈ (1(,)2)((cos‘𝑝) = 0 ∧ ∀𝑥 ∈ (𝑝(,)(2 · 𝑝))0 < (sin‘𝑥))
 
7-Mar-2024cosz12 13495 Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
∃𝑝 ∈ (1(,)2)(cos‘𝑝) = 0
 
6-Mar-2024cos12dec 11730 Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
((𝐴 ∈ (1[,]2) ∧ 𝐵 ∈ (1[,]2) ∧ 𝐴 < 𝐵) → (cos‘𝐵) < (cos‘𝐴))
 
2-Mar-2024plusffvalg 12616 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
 
25-Feb-2024insubm 12703 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))
 
25-Feb-2024mul2lt0pn 9721 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐵 · 𝐴) < 0)
 
25-Feb-2024mul2lt0np 9720 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐴 · 𝐵) < 0)
 
25-Feb-2024lt0ap0 8567 A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 # 0)
 
25-Feb-2024negap0d 8550 The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → -𝐴 # 0)
 
24-Feb-2024lt0ap0d 8568 A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    ⇒   (𝜑 → 𝐴 # 0)
 
20-Feb-2024ivthdec 13416 The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) < (𝐹‘𝑥))    ⇒   (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
20-Feb-2024ivthinclemex 13414 Lemma for ivthinc 13415. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑧 ∧ ∀𝑟 ∈ 𝑅 𝑧 < 𝑟))
 
19-Feb-2024ivthinclemuopn 13410 Lemma for ivthinc 13415. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    &   (𝜑 → 𝑆 ∈ 𝑅)    ⇒   (𝜑 → ∃𝑞 ∈ 𝑅 𝑞 < 𝑆)
 
19-Feb-2024dedekindicc 13405 A Dedekind cut identifies a unique real number. Similar to df-inp 7428 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃!𝑥 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
19-Feb-2024grpsubfvalg 12748 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝐼 = (invg‘𝐺)    &    − = (-g‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
 
18-Feb-2024ivthinclemloc 13413 Lemma for ivthinc 13415. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑅)))
 
18-Feb-2024ivthinclemdisj 13412 Lemma for ivthinc 13415. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → (𝐿 ∩ 𝑅) = ∅)
 
18-Feb-2024ivthinclemur 13411 Lemma for ivthinc 13415. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑅 ↔ ∃𝑞 ∈ 𝑅 𝑞 < 𝑟))
 
18-Feb-2024ivthinclemlr 13409 Lemma for ivthinc 13415. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
 
18-Feb-2024ivthinclemum 13407 Lemma for ivthinc 13415. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅)
 
18-Feb-2024ivthinclemlm 13406 Lemma for ivthinc 13415. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
 
17-Feb-20240subm 12702 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
17-Feb-2024mndissubm 12697 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
 
17-Feb-2024mgmsscl 12615 If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    ⇒   (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
 
15-Feb-2024dedekindicclemeu 13403 Lemma for dedekindicc 13405. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟))    &   (𝜑 → 𝐷 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟))    &   (𝜑 → 𝐶 < 𝐷)    ⇒   (𝜑 → ⊥)
 
15-Feb-2024dedekindicclemlu 13402 Lemma for dedekindicc 13405. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
15-Feb-2024dedekindicclemlub 13401 Lemma for dedekindicc 13405. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)))
 
15-Feb-2024dedekindicclemloc 13400 Lemma for dedekindicc 13405. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦)))
 
15-Feb-2024dedekindicclemub 13399 Lemma for dedekindicc 13405. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
 
15-Feb-2024dedekindicclemuub 13398 Lemma for dedekindicc 13405. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐶)
 
14-Feb-2024suplociccex 13397 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7992 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐵 < 𝐶)    &   (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐵[,]𝐶)(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐵[,]𝐶)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
14-Feb-2024suplociccreex 13396 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7992 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐵 < 𝐶)    &   (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
6-Feb-2024ivthinclemlopn 13408 Lemma for ivthinc 13415. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    &   (𝜑 → 𝑄 ∈ 𝐿)    ⇒   (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)
 
5-Feb-2024ivthinc 13415 The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    ⇒   (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
2-Feb-2024dedekindeulemuub 13389 Lemma for dedekindeu 13395. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 ∈ 𝑈)    ⇒   (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐴)
 
31-Jan-2024dedekindeulemeu 13394 Lemma for dedekindeu 13395. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐴 ∧ ∀𝑟 ∈ 𝑈 𝐴 < 𝑟))    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐵 ∧ ∀𝑟 ∈ 𝑈 𝐵 < 𝑟))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ⊥)
 
31-Jan-2024dedekindeulemlu 13393 Lemma for dedekindeu 13395. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
31-Jan-2024dedekindeulemlub 13392 Lemma for dedekindeu 13395. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)))
 
31-Jan-2024dedekindeulemloc 13391 Lemma for dedekindeu 13395. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦)))
 
31-Jan-2024dedekindeulemub 13390 Lemma for dedekindeu 13395. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
 
30-Jan-2024axsuploc 7992 An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7895 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
30-Jan-2024iotam 5190 Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is inhabited (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
29-Jan-2024sgrpidmndm 12656 A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
 
24-Jan-2024axpre-suploclemres 7863 Lemma for axpre-suploc 7864. The result. The proof just needs to define 𝐵 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ℝ), and apply suplocsr 7771. (Contributed by Jim Kingdon, 24-Jan-2024.)
(𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)    &   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))    &   𝐵 = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024ax-pre-suploc 7895 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound.

Although this and ax-caucvg 7894 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7894.

(Contributed by Jim Kingdon, 23-Jan-2024.)

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024axpre-suploc 7864 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given 𝑥 < 𝑦, either there is an element of the set greater than 𝑥, or 𝑦 is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7895. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
22-Jan-2024suplocsr 7771 An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
 
21-Jan-2024bj-el2oss1o 13809 Shorter proof of el2oss1o 6422 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o → 𝐴 ⊆ 1o)
 
21-Jan-2024ltm1sr 7739 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴 ∈ R → (𝐴 +R -1R) <R 𝐴)
 
20-Jan-2024mndinvmod 12678 Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 
19-Jan-2024suplocsrlempr 7769 Lemma for suplocsr 7771. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)))
 
18-Jan-2024suplocsrlemb 7768 Lemma for suplocsr 7771. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∀𝑢 ∈ P ∀𝑣 ∈ P (𝑢<P 𝑣 → (∃𝑞 ∈ 𝐵 𝑢<P 𝑞 ∨ ∀𝑞 ∈ 𝐵 𝑞<P 𝑣)))
 
16-Jan-2024suplocsrlem 7770 Lemma for suplocsr 7771. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
 
14-Jan-2024suplocexprlemlub 7686 Lemma for suplocexpr 7687. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
 
14-Jan-2024suplocexprlemub 7685 Lemma for suplocexpr 7687. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)
 
10-Jan-2024nfcsbw 3085 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3086 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
 
10-Jan-2024nfsbcdw 3083 Version of nfsbcd 2974 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvcsbw 3053 Version of cbvcsb 3054 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝐶    &   Ⅎ𝑥𝐷    &   (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
 
10-Jan-2024cbvsbcw 2982 Version of cbvsbc 2983 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvrex2vw 2708 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2710 with a disjoint variable condition, which does not require ax-13 2143. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvral2vw 2707 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2709 with a disjoint variable condition, which does not require ax-13 2143. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvralw 2691 Rule used to change bound variables, using implicit substitution. Version of cbvral 2692 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1500 and ax-bndl 1502 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexfw 2688 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2690 with a disjoint variable condition, which does not require ax-13 2143. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfralw 2507 Bound-variable hypothesis builder for restricted quantification. See nfralya 2510 for a version with 𝑦 and 𝐴 distinct instead of 𝑥 and 𝑦. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfraldw 2502 Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2505 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfabdw 2331 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2332 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
10-Jan-2024cbv2w 1743 Rule used to change bound variables, using implicit substitution. Version of cbv2 1742 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
9-Jan-2024suplocexprlemloc 7683 Lemma for suplocexpr 7687. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
 
9-Jan-2024suplocexprlemdisj 7682 Lemma for suplocexpr 7687. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
 
9-Jan-2024suplocexprlemru 7681 Lemma for suplocexpr 7687. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
 
9-Jan-2024suplocexprlemrl 7679 Lemma for suplocexpr 7687. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
 
9-Jan-2024suplocexprlem2b 7676 Lemma for suplocexpr 7687. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
 
9-Jan-2024suplocexprlemell 7675 Lemma for suplocexpr 7687. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))
 
7-Jan-2024suplocexpr 7687 An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
 
7-Jan-2024suplocexprlemex 7684 Lemma for suplocexpr 7687. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → 𝐵 ∈ P)
 
7-Jan-2024suplocexprlemmu 7680 Lemma for suplocexpr 7687. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))
 
7-Jan-2024suplocexprlemml 7678 Lemma for suplocexpr 7687. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))
 
7-Jan-2024suplocexprlemss 7677 Lemma for suplocexpr 7687. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → 𝐴 ⊆ P)
 
5-Jan-2024dedekindicclemicc 13404 Lemma for dedekindicc 13405. Same as dedekindicc 13405, except that we merely show 𝑥 to be an element of (𝐴[,]𝐵). Later we will strengthen that to (𝐴(,)𝐵). (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃!𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
5-Jan-2024dedekindeu 13395 A Dedekind cut identifies a unique real number. Similar to df-inp 7428 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃!𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
31-Dec-2023dvmptsubcn 13479 Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ 𝑊)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷))    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷)))
 
31-Dec-2023dvmptnegcn 13478 Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ -𝐴)) = (𝑥 ∈ ℂ ↦ -𝐵))
 
31-Dec-2023dvmptcmulcn 13477 Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   (𝜑 → 𝐶 ∈ ℂ)    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐶 · 𝐴))) = (𝑥 ∈ ℂ ↦ (𝐶 · 𝐵)))
 
31-Dec-2023brm 4039 If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)
 
30-Dec-2023dvmptccn 13473 Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0))
 
30-Dec-2023dvmptidcn 13472 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)
 
29-Dec-2023mndbn0 12667 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12666). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
26-Dec-2023lidrididd 12636 If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 12635) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    ⇒   (𝜑 → 𝐿 = 0 )
 
26-Dec-2023lidrideqd 12635 If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    ⇒   (𝜑 → 𝐿 = 𝑅)
 
25-Dec-2023ctfoex 7095 A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)
 
23-Dec-2023enct 12388 Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
23-Dec-2023enctlem 12387 Lemma for enct 12388. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
23-Dec-2023omct 7094 ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
 
21-Dec-2023dvcoapbr 13465 The chain rule for derivatives at a point. The 𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑌⟶𝑋)    &   (𝜑 → 𝑌 ⊆ 𝑇)    &   (𝜑 → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝑇 ⊆ ℂ)    &   (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))
 
19-Dec-2023apsscn 8566 The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
{𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ
 
19-Dec-2023aprcl 8565 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
 
18-Dec-2023limccoap 13441 Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})    &   ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}) → 𝑆 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) limℂ 𝑋))    &   (𝜑 → 𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) limℂ 𝐶))    &   (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   (𝜑 → 𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) limℂ 𝑋))
 
16-Dec-2023cnreim 10942 Complex apartness in terms of real and imaginary parts. See also apreim 8522 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ ((ℜ‘𝐴) # (ℜ‘𝐵) ∨ (ℑ‘𝐴) # (ℑ‘𝐵))))
 
14-Dec-2023cnopnap 13388 The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 𝐴} ∈ (MetOpen‘(abs ∘ − )))
 
14-Dec-2023cnovex 12990 The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
 
13-Dec-2023reopnap 13332 The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ → {𝑤 ∈ ℝ ∣ 𝑤 # 𝐴} ∈ (topGen‘ran (,)))
 
12-Dec-2023cnopncntop 13331 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ ∈ (MetOpen‘(abs ∘ − ))
 
12-Dec-2023unicntopcntop 13330 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ = ∪ (MetOpen‘(abs ∘ − ))
 
4-Dec-2023bj-pm2.18st 13785 Clavius law for stable formulas. See pm2.18dc 850. (Contributed by BJ, 4-Dec-2023.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
 
4-Dec-2023bj-nnclavius 13772 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
 
2-Dec-2023dvmulxx 13462 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 13460. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))    ⇒   (𝜑 → ((𝑆 D (𝐹 ∘𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
 
1-Dec-2023dvmulxxbr 13460 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 13462. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶))))
 
29-Nov-2023subctctexmid 14034 If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))    &   (𝜑 → ω ∈ Markov)    ⇒   (𝜑 → EXMID)
 
29-Nov-2023ismkvnex 7131 The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
 
28-Nov-2023exmid1stab 14033 If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})    ⇒   (𝜑 → EXMID)
 
28-Nov-2023ccfunen 7226 Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ≈ ω)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
 
27-Nov-2023df-cc 7225 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7183 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)))
 
26-Nov-2023offeq 6074 Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑 → 𝐹:𝐴⟶𝑆)    &   (𝜑 → 𝐺:𝐵⟶𝑇)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝐴 ∩ 𝐵) = 𝐶    &   (𝜑 → 𝐻:𝐶⟶𝑈)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)    &   ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥))    ⇒   (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)
 
25-Nov-2023dvaddxx 13461 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 13459. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))    ⇒   (𝜑 → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))
 
25-Nov-2023dvaddxxbr 13459 The sum rule for derivatives at a point. That is, if the derivative of 𝐹 at 𝐶 is 𝐾 and the derivative of 𝐺 at 𝐶 is 𝐿, then the derivative of the pointwise sum of those two functions at 𝐶 is 𝐾 + 𝐿. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(𝐾 + 𝐿))
 
25-Nov-2023dcnn 843 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 837. The relation between dcn 837 and dcnn 843 is analogous to that between notnot 624 and notnotnot 629 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 843 means that a proposition is testable if and only if its negation is testable, and dcn 837 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
(DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)
 
24-Nov-2023bj-dcst 13796 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB 𝜑 ↔ STAB 𝜑)
 
24-Nov-2023bj-nnbidc 13792 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 13779. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-dcstab 13791 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-fadc 13789 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → DECID 𝜑)
 
24-Nov-2023bj-trdc 13787 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → DECID 𝜑)
 
24-Nov-2023bj-stal 13784 The universal quantification of a stable formula is stable. See bj-stim 13781 for implication, stabnot 828 for negation, and bj-stan 13782 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)
 
24-Nov-2023bj-stand 13783 The conjunction of two stable formulas is stable. Deduction form of bj-stan 13782. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 13782 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → STAB 𝜓)    &   (𝜑 → STAB 𝜒)    ⇒   (𝜑 → STAB (𝜓 ∧ 𝜒))
 
24-Nov-2023bj-stan 13782 The conjunction of two stable formulas is stable. See bj-stim 13781 for implication, stabnot 828 for negation, and bj-stal 13784 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))
 
24-Nov-2023bj-stim 13781 A conjunction with a stable consequent is stable. See stabnot 828 for negation , bj-stan 13782 for conjunction , and bj-stal 13784 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB 𝜓 → STAB (𝜑 → 𝜓))
 
24-Nov-2023bj-nnbist 13779 If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 13792). (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-fast 13776 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-trst 13774 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → STAB 𝜑)
 
24-Nov-2023bj-nnan 13771 The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
 
24-Nov-2023bj-nnim 13770 The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))
 
24-Nov-2023bj-nnsn 13768 As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))
 
24-Nov-2023nnal 1642 The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
 
22-Nov-2023ofvalg 6070 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝐴 ∩ 𝐵) = 𝑆    &   ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    &   ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 
21-Nov-2023exmidac 7186 The axiom of choice implies excluded middle. See acexmid 5852 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
(CHOICE → EXMID)
 
21-Nov-2023exmidaclem 7185 Lemma for exmidac 7186. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}    &   𝐶 = {𝐴, 𝐵}    ⇒   (CHOICE → EXMID)
 
21-Nov-2023exmid1dc 4186 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4179 or ordtriexmid 4505. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})    ⇒   (𝜑 → EXMID)
 
20-Nov-2023acfun 7184 A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
(𝜑 → CHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
 
18-Nov-2023condc 848 Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

(DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
18-Nov-2023stdcn 842 A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 837. (Contributed by BJ, 18-Nov-2023.)
(STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑))
 
17-Nov-2023cnplimclemr 13432 Lemma for cnplimccntop 13433. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐴)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    ⇒   (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
 
17-Nov-2023cnplimclemle 13431 Lemma for cnplimccntop 13433. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐴)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    &   (𝜑 → 𝐸 ∈ ℝ+)    &   (𝜑 → 𝐷 ∈ ℝ+)    &   (𝜑 → 𝑍 ∈ 𝐴)    &   ((𝜑 ∧ 𝑍 # 𝐵 ∧ (abs‘(𝑍 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < (𝐸 / 2))    &   (𝜑 → (abs‘(𝑍 − 𝐵)) < 𝐷)    ⇒   (𝜑 → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < 𝐸)
 
14-Nov-2023limccnp2cntop 13440 The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   (𝜑 → 𝑋 ⊆ ℂ)    &   (𝜑 → 𝑌 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    ⇒   (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵))
 
10-Nov-2023rpmaxcl 11187 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)
 
9-Nov-2023limccnp2lem 13439 Lemma for limccnp2cntop 13440. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   (𝜑 → 𝑋 ⊆ ℂ)    &   (𝜑 → 𝑌 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    &   Ⅎ𝑥𝜑    &   (𝜑 → 𝐸 ∈ ℝ+)    &   (𝜑 → 𝐿 ∈ ℝ+)    &   (𝜑 → ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝐿 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝐿) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝐸))    &   (𝜑 → 𝐹 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐹) → (abs‘(𝑅 − 𝐶)) < 𝐿))    &   (𝜑 → 𝐺 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐺) → (abs‘(𝑆 − 𝐷)) < 𝐿))    ⇒   (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝐸))
 
4-Nov-2023ellimc3apf 13423 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   Ⅎ𝑧𝐹    ⇒   (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
 
3-Nov-2023limcmpted 13426 Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
(𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℂ)    ⇒   (𝜑 → (𝐶 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘(𝐷 − 𝐶)) < 𝑥))))
 
1-Nov-2023unct 12397 The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
((∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) → ∃ℎ ℎ:ω–onto→((𝐴 ∪ 𝐵) ⊔ 1o))
 
31-Oct-2023ctiunct 12395 A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each 𝐵(𝑥): it refers to 𝐵(𝑥) together with the 𝐺(𝑥) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be (𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑔𝑔:ω–onto→(𝐵 ⊔ 1o). This is almost omiunct 12399 (which uses countable choice) although that is for a countably infinite collection not any countable collection.

Compare with the case of two sets instead of countably many, as seen at unct 12397, which says that the union of two countable sets is countable .

The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12350) and using the first number to map to an element 𝑥 of 𝐴 and the second number to map to an element of B(x) . In this way we are able to map to every element of ∪ 𝑥 ∈ 𝐴𝐵. Although it would be possible to work directly with countability expressed as 𝐹:ω–onto→(𝐴 ⊔ 1o), we instead use functions from subsets of the natural numbers via ctssdccl 7088 and ctssdc 7090.

(Contributed by Jim Kingdon, 31-Oct-2023.)

(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
 
30-Oct-2023ctssdccl 7088 A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7090 but expressed in terms of classes rather than ∃. (Contributed by Jim Kingdon, 30-Oct-2023.)
(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   𝑆 = {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)}    &   𝐺 = (◡inl ∘ 𝐹)    ⇒   (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))
 
28-Oct-2023ctiunctlemfo 12394 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2nd ‘(𝐽‘𝑛))))    &   Ⅎ𝑥𝐻    &   Ⅎ𝑥𝑈    ⇒   (𝜑 → 𝐻:𝑈–onto→∪ 𝑥 ∈ 𝐴 𝐵)
 
28-Oct-2023ctiunctlemf 12393 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2nd ‘(𝐽‘𝑛))))    ⇒   (𝜑 → 𝐻:𝑈⟶∪ 𝑥 ∈ 𝐴 𝐵)
 
28-Oct-2023ctiunctlemudc 12392 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    ⇒   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑈)
 
28-Oct-2023ctiunctlemuom 12391 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    ⇒   (𝜑 → 𝑈 ⊆ ω)
 
28-Oct-2023ctiunctlemu2nd 12390 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   (𝜑 → 𝑁 ∈ 𝑈)    ⇒   (𝜑 → (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)
 
28-Oct-2023ctiunctlemu1st 12389 Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   (𝜑 → 𝑁 ∈ 𝑈)    ⇒   (𝜑 → (1st ‘(𝐽‘𝑁)) ∈ 𝑆)
 
28-Oct-2023pm2.521gdc 863 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
(DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜒 → 𝜑)))
 
28-Oct-2023stdcndc 840 A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)
 
28-Oct-2023conax1k 649 Weakening of conax1 648. General instance of pm2.51 650 and of pm2.52 651. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → (𝜒 → ¬ 𝜓))
 
28-Oct-2023conax1 648 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → ¬ 𝜓)
 
25-Oct-2023divcnap 13349 Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   𝐾 = (𝐽 ↾t {𝑥 ∈ ℂ ∣ 𝑥 # 0})    ⇒   (𝑦 ∈ ℂ, 𝑧 ∈ {𝑥 ∈ ℂ ∣ 𝑥 # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
 
23-Oct-2023cnm 7794 A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ∃𝑥 𝑥 ∈ 𝐴)
 
23-Oct-2023oprssdmm 6150 Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
((𝜑 ∧ 𝑢 ∈ 𝑆) → ∃𝑣 𝑣 ∈ 𝑢)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   (𝜑 → Rel 𝐹)    ⇒   (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
 
22-Oct-2023addcncntoplem 13345 Lemma for addcncntop 13346, subcncntop 13347, and mulcncntop 13348. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &    + :(ℂ × ℂ)⟶ℂ    &   ((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑦 ∧ (abs‘(𝑣 − 𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎))    ⇒    + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
22-Oct-2023txmetcnp 13312 Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)    ⇒   (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
 
22-Oct-2023xmetxpbl 13302 The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐶 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   (𝜑 → 𝑅 ∈ ℝ*)    &   (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))    ⇒   (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1st ‘𝐶)(ball‘𝑀)𝑅) × ((2nd ‘𝐶)(ball‘𝑁)𝑅)))
 
15-Oct-2023xmettxlem 13303 Lemma for xmettx 13304. (Contributed by Jim Kingdon, 15-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)    ⇒   (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))
 
11-Oct-2023xmettx 13304 The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)    ⇒   (𝜑 → 𝐿 = (𝐽 ×t 𝐾))
 
11-Oct-2023xmetxp 13301 The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    ⇒   (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
 
29-Sep-2023syl2anc2 410 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(𝜑 → 𝜓)    &   (𝜓 → 𝜒)    &   ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   (𝜑 → 𝜃)
 
12-Sep-2023pwntru 4185 A slight strengthening of pwtrufal 14030. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
((𝐴 ⊆ {∅} ∧ 𝐴 ≠ {∅}) → 𝐴 = ∅)
 
11-Sep-2023pwtrufal 14030 A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4184. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4182), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
(𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))
 
9-Sep-2023mathbox 13767 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

𝜑    ⇒   𝜑
 
6-Sep-2023djuexb 7021 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V)
 
3-Sep-2023pwf1oexmid 14032 An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)    ⇒   ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2o ∧ EXMID)))
 
3-Sep-2023pwle2 14031 An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)    ⇒   ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o)
 
30-Aug-2023isomninn 14063 Omniscience stated in terms of natural numbers. Similar to isomnimap 7113 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))
 
30-Aug-2023isomninnlem 14062 Lemma for isomninn 14063. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))
 
28-Aug-2023trilpolemisumle 14070 Lemma for trilpo 14075. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℕ)    ⇒   (𝜑 → Σ𝑖 ∈ 𝑍 ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ 𝑍 (1 / (2↑𝑖)))
 
25-Aug-2023cvgcmp2n 14065 A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))    ⇒   (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
 
25-Aug-2023cvgcmp2nlemabs 14064 Lemma for cvgcmp2n 14065. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)‘𝑁) as the sum of (seq1( + , 𝐺)‘𝑀) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))
 
24-Aug-2023trilpolemclim 14068 Lemma for trilpo 14075. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐺 = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))    ⇒   (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
 
23-Aug-2023trilpo 14075 Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 14073 (which means the sequence contains a zero), trilpolemeq1 14072 (which means the sequence is all ones), and trilpolemgt1 14071 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14061) or that the real numbers are a discrete field (see trirec0 14076).

LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10199 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ω ∈ Omni)
 
23-Aug-2023trilpolemres 14074 Lemma for trilpo 14075. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   (𝜑 → (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴))    ⇒   (𝜑 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1))
 
23-Aug-2023trilpolemlt1 14073 Lemma for trilpo 14075. The 𝐴 < 1 case. We can use the distance between 𝐴 and one (that is, 1 − 𝐴) to find a position in the sequence 𝑛 where terms after that point will not add up to as much as 1 − 𝐴. By finomni 7116 we know the terms up to 𝑛 either contain a zero or are all one. But if they are all one that contradicts the way we constructed 𝑛, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   (𝜑 → 𝐴 < 1)    ⇒   (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
 
23-Aug-2023trilpolemeq1 14072 Lemma for trilpo 14075. The 𝐴 = 1 case. This is proved by noting that if any (𝐹‘𝑥) is zero, then the infinite sum 𝐴 is less than one based on the term which is zero. We are using the fact that the 𝐹 sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   (𝜑 → 𝐴 = 1)    ⇒   (𝜑 → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)
 
23-Aug-2023trilpolemgt1 14071 Lemma for trilpo 14075. The 1 < 𝐴 case. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    ⇒   (𝜑 → ¬ 1 < 𝐴)
 
23-Aug-2023trilpolemcl 14069 Lemma for trilpo 14075. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    ⇒   (𝜑 → 𝐴 ∈ ℝ)
 
23-Aug-2023triap 14061 Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ DECID 𝐴 # 𝐵))
 
19-Aug-2023djuenun 7189 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ∪ 𝐷))
 
16-Aug-2023ctssdclemr 7089 Lemma for ctssdc 7090. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))
 
16-Aug-2023ctssdclemn0 7087 Lemma for ctssdc 7090. The ¬ ∅ ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   (𝜑 → ¬ ∅ ∈ 𝑆)    ⇒   (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
 
15-Aug-2023ctssexmid 7126 The decidability condition in ctssdc 7090 is needed. More specifically, ctssdc 7090 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o))    &   ω ∈ Omni    ⇒   (𝜑 ∨ ¬ 𝜑)
 
15-Aug-2023ctssdc 7090 A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7126. (Contributed by Jim Kingdon, 15-Aug-2023.)
(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠) ↔ ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
14-Aug-2023mpoexw 6192 Weak version of mpoex 6193 that holds without ax-coll 4104. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐷 ∈ V    &   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷    ⇒   (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
 
13-Aug-2023grpinvfvalg 12745 The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
 
13-Aug-2023ltntri 8047 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))
 
13-Aug-2023mptexw 6092 Weak version of mptex 5722 that holds without ax-coll 4104. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝐴 ∈ V    &   𝐶 ∈ V    &   ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
 
13-Aug-2023funexw 6091 Weak version of funex 5719 that holds without ax-coll 4104. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V)
 
11-Aug-2023qnnen 12386 The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.)
ℚ ≈ ℕ
 
10-Aug-2023ctinfomlemom 12382 Lemma for ctinfom 12383. Converting between ω and ℕ0. (Contributed by Jim Kingdon, 10-Aug-2023.)
𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐺 = (𝐹 ∘ ◡𝑁)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹‘𝑘) ∈ (𝐹 “ 𝑛))    ⇒   (𝜑 → (𝐺:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝐺‘𝑗) ≠ (𝐺‘𝑖)))
 
9-Aug-2023difinfsnlem 7076 Lemma for difinfsn 7077. The case where we need to swap 𝐵 and (inr‘∅) in building the mapping 𝐺. (Contributed by Jim Kingdon, 9-Aug-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → 𝐹:(ω ⊔ 1o)–1-1→𝐴)    &   (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)    &   𝐺 = (𝑛 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))    ⇒   (𝜑 → 𝐺:ω–1-1→(𝐴 ∖ {𝐵}))
 
8-Aug-2023difinfinf 7078 An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
(((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵))
 
8-Aug-2023difinfsn 7077 An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ (𝐴 ∖ {𝐵}))
 
7-Aug-2023ctinf 12385 A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.)
(𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))
 
7-Aug-2023inffinp1 12384 An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → ω ≼ 𝐴)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
 
7-Aug-2023ctinfom 12383 A condition for a set being countably infinite. Restates ennnfone 12380 in terms of ω and function image. Like ennnfone 12380 the condition can be summarized as 𝐴 being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
(𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
 
6-Aug-2023rerestcntop 13344 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   𝑅 = (topGen‘ran (,))    ⇒   (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴))
 
6-Aug-2023tgioo2cntop 13343 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (topGen‘ran (,)) = (𝐽 ↾t ℝ)
 
4-Aug-2023nninffeq 14053 Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, (𝜑 → ∀𝑛 ∈ suc ω...). (Contributed by Jim Kingdon, 4-Aug-2023.)
(𝜑 → 𝐹:ℕ∞⟶ℕ0)    &   (𝜑 → 𝐺:ℕ∞⟶ℕ0)    &   (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o)))    &   (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
3-Aug-2023txvalex 13048 Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 13054. (Contributed by Jim Kingdon, 3-Aug-2023.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)
 
3-Aug-2023hashfingrpnn 12739 A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → (♯‘𝐵) ∈ ℕ)
 
3-Aug-2023hashfinmndnn 12668 A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → (♯‘𝐵) ∈ ℕ)
 
3-Aug-2023dvdsgcdidd 11949 The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑀 ∥ 𝑁)    ⇒   (𝜑 → (𝑀 gcd 𝑁) = 𝑀)
 
3-Aug-2023gcdmultipled 11948 The greatest common divisor of a nonnegative integer 𝑀 and a multiple of it is 𝑀 itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀)
 
3-Aug-2023fihashelne0d 10732 A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → ¬ (♯‘𝐴) = 0)
 
3-Aug-2023phpeqd 6910 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6843 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐴 ≈ 𝐵)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
3-Aug-2023enpr2d 6795 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
3-Aug-2023elrnmpt2d 4866 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝜑 → 𝐶 ∈ ran 𝐹)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
 
3-Aug-2023elrnmptdv 4865 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)    ⇒   (𝜑 → 𝐷 ∈ ran 𝐹)
 
3-Aug-2023rspcime 2841 Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
3-Aug-2023neqcomd 2175 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → ¬ 𝐴 = 𝐵)    ⇒   (𝜑 → ¬ 𝐵 = 𝐴)
 
2-Aug-2023dvid 13456 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(ℂ D ( I ↾ ℂ)) = (ℂ × {1})
 
2-Aug-2023dvconst 13455 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0}))
 
2-Aug-2023dvidlemap 13454 Lemma for dvid 13456 and dvconst 13455. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(𝜑 → 𝐹:ℂ⟶ℂ)    &   ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   𝐵 ∈ ℂ    ⇒   (𝜑 → (ℂ D 𝐹) = (ℂ × {𝐵}))
 
2-Aug-2023diveqap1bd 8753 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8622. Converse of diveqap1d 8715. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝐴 / 𝐵) = 1)
 
31-Jul-2023mul0inf 11204 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11026 and mulap0bd 8575 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ inf({(abs‘𝐴), (abs‘𝐵)}, ℝ, < ) = 0))
 
31-Jul-2023mul0eqap 8588 If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 𝐵)    &   (𝜑 → (𝐴 · 𝐵) = 0)    ⇒   (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0))
 
31-Jul-2023apcon4bid 8543 Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    &   (𝜑 → (𝐴 # 𝐵 ↔ 𝐶 # 𝐷))    ⇒   (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
 
30-Jul-2023uzennn 10392 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝑀 ∈ ℤ → (ℤ≥‘𝑀) ≈ ℕ)
 
30-Jul-2023djuen 7188 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))
 
30-Jul-2023endjudisj 7187 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
 
30-Jul-2023eninr 7075 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴)
 
30-Jul-2023eninl 7074 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
 
29-Jul-2023exmidunben 12381 If any unbounded set of positive integers is equinumerous to ℕ, then the Limited Principle of Omniscience (LPO) implies excluded middle. (Contributed by Jim Kingdon, 29-Jul-2023.)
((∀𝑥((𝑥 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝑥 𝑚 < 𝑛) → 𝑥 ≈ ℕ) ∧ ω ∈ Omni) → EXMID)
 
29-Jul-2023exmidsssnc 4189 Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4184 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4188 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
(𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
 
28-Jul-2023dvfcnpm 13453 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
(𝐹 ∈ (ℂ ↑pm ℂ) → (ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ)
 
28-Jul-2023dvfpm 13452 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
(𝐹 ∈ (ℂ ↑pm ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ)
 
23-Jul-2023ennnfonelemhdmp1 12364 Lemma for ennnfone 12380. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    &   (𝜑 → ¬ (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)))    ⇒   (𝜑 → dom (𝐻‘(𝑃 + 1)) = suc dom (𝐻‘𝑃))
 
23-Jul-2023ennnfonelemp1 12361 Lemma for ennnfone 12380. Value of 𝐻 at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → (𝐻‘(𝑃 + 1)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
 
22-Jul-2023nntr2 6482 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
 
22-Jul-2023nnsssuc 6481 A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
 
21-Jul-2023ennnfoneleminc 12366 Lemma for ennnfone 12380. We only add elements to 𝐻 as the index increases. (Contributed by Jim Kingdon, 21-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    &   (𝜑 → 𝑄 ∈ ℕ0)    &   (𝜑 → 𝑃 ≤ 𝑄)    ⇒   (𝜑 → (𝐻‘𝑃) ⊆ (𝐻‘𝑄))
 
20-Jul-2023ennnfonelemg 12358 Lemma for ennnfone 12380. Closure for 𝐺. (Contributed by Jim Kingdon, 20-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
 
20-Jul-2023ennnfonelemjn 12357 Lemma for ennnfone 12380. Non-initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
 
20-Jul-2023ennnfonelemj0 12356 Lemma for ennnfone 12380. Initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
 
20-Jul-2023seqp1cd 10422 Value of the sequence builder function at a successor. A version of seq3p1 10418 which provides two classes 𝐷 and 𝐶 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
20-Jul-2023seqovcd 10419 A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10420 and seq1cd 10421 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    ⇒   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶)
 
19-Jul-2023ennnfonelemhom 12370 Lemma for ennnfone 12380. The sequences in 𝐻 increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑀 ∈ ω)    ⇒   (𝜑 → ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖))
 
19-Jul-2023ennnfonelemex 12369 Lemma for ennnfone 12380. Extending the sequence (𝐻‘𝑃) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → ∃𝑖 ∈ ℕ0 dom (𝐻‘𝑃) ∈ dom (𝐻‘𝑖))
 
19-Jul-2023ennnfonelemkh 12367 Lemma for ennnfone 12380. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → dom (𝐻‘𝑃) ⊆ (◡𝑁‘𝑃))
 
19-Jul-2023ennnfonelemom 12363 Lemma for ennnfone 12380. 𝐻 yields finite sequences. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → dom (𝐻‘𝑃) ∈ ω)
 
19-Jul-2023ennnfonelem1 12362 Lemma for ennnfone 12380. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   (𝜑 → (𝐻‘1) = {⟨∅, (𝐹‘∅)⟩})
 
19-Jul-2023seq1cd 10421 Initial value of the recursive sequence builder. A version of seq3-1 10416 which provides two classes 𝐷 and 𝐶 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
 
17-Jul-2023ennnfonelemhf1o 12368 Lemma for ennnfone 12380. Each of the functions in 𝐻 is one to one and onto an image of 𝐹. (Contributed by Jim Kingdon, 17-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → (𝐻‘𝑃):dom (𝐻‘𝑃)–1-1-onto→(𝐹 “ (◡𝑁‘𝑃)))
 
16-Jul-2023ennnfonelemen 12376 Lemma for ennnfone 12380. The result. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)    ⇒   (𝜑 → 𝐴 ≈ ℕ)
 
16-Jul-2023ennnfonelemdm 12375 Lemma for ennnfone 12380. The function 𝐿 is defined everywhere. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)    ⇒   (𝜑 → dom 𝐿 = ω)
 
16-Jul-2023ennnfonelemrn 12374 Lemma for ennnfone 12380. 𝐿 is onto 𝐴. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)    ⇒   (𝜑 → ran 𝐿 = 𝐴)
 
16-Jul-2023ennnfonelemf1 12373 Lemma for ennnfone 12380. 𝐿 is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)    ⇒   (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
 
16-Jul-2023ennnfonelemfun 12372 Lemma for ennnfone 12380. 𝐿 is a function. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)    ⇒   (𝜑 → Fun 𝐿)
 
16-Jul-2023ennnfonelemrnh 12371 Lemma for ennnfone 12380. A consequence of ennnfonelemss 12365. (Contributed by Jim Kingdon, 16-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑋 ∈ ran 𝐻)    &   (𝜑 → 𝑌 ∈ ran 𝐻)    ⇒   (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
 
15-Jul-2023ennnfonelemss 12365 Lemma for ennnfone 12380. We only add elements to 𝐻 as the index increases. (Contributed by Jim Kingdon, 15-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    &   (𝜑 → 𝑃 ∈ ℕ0)    ⇒   (𝜑 → (𝐻‘𝑃) ⊆ (𝐻‘(𝑃 + 1)))
 
15-Jul-2023ennnfonelem0 12360 Lemma for ennnfone 12380. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   (𝜑 → (𝐻‘0) = ∅)
 
15-Jul-2023ennnfonelemk 12355 Lemma for ennnfone 12380. (Contributed by Jim Kingdon, 15-Jul-2023.)
(𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → 𝐾 ∈ ω)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → ∀𝑗 ∈ suc 𝑁(𝐹‘𝐾) ≠ (𝐹‘𝑗))    ⇒   (𝜑 → 𝑁 ∈ 𝐾)
 
15-Jul-2023ennnfonelemdc 12354 Lemma for ennnfone 12380. A direct consequence of fidcenumlemrk 6931. (Contributed by Jim Kingdon, 15-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → 𝑃 ∈ ω)    ⇒   (𝜑 → DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))
 
14-Jul-2023djur 7046 A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
(𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
 
13-Jul-2023sbthomlem 14057 Lemma for sbthom 14058. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
(𝜑 → ω ∈ Omni)    &   (𝜑 → 𝑌 ⊆ {∅})    &   (𝜑 → 𝐹:ω–1-1-onto→(𝑌 ⊔ ω))    ⇒   (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅}))
 
12-Jul-2023caseinr 7069 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
(𝜑 → Fun 𝐹)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺‘𝐴))
 
12-Jul-2023inl11 7042 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
 
11-Jul-2023djudomr 7197 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≼ (𝐴 ⊔ 𝐵))
 
11-Jul-2023djudoml 7196 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵))
 
11-Jul-2023omp1eomlem 7071 Lemma for omp1eom 7072. (Contributed by Jim Kingdon, 11-Jul-2023.)
𝐹 = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))    &   𝑆 = (𝑥 ∈ ω ↦ suc 𝑥)    &   𝐺 = case(𝑆, ( I ↾ 1o))    ⇒   𝐹:ω–1-1-onto→(ω ⊔ 1o)
 
11-Jul-2023xp01disjl 6413 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
(({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
 
10-Jul-2023sbthom 14058 Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 14056 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ω? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID)
 
10-Jul-2023endjusym 7073 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))
 
10-Jul-2023omp1eom 7072 Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
(ω ⊔ 1o) ≈ ω
 
9-Jul-2023refeq 14060 Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
(𝜑 → 𝐹:ℝ⟶ℝ)    &   (𝜑 → 𝐺:ℝ⟶ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥)))    &   (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
9-Jul-2023seqvalcd 10415 Value of the sequence builder function. Similar to seq3val 10414 but the classes 𝐷 (type of each term) and 𝐶 (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
(𝜑 → 𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)    &   (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
 
9-Jul-2023djuun 7044 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
 
9-Jul-2023djuin 7041 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
 
8-Jul-2023limcimo 13428 Conditions which ensure there is at most one limit value of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ (𝐾 ↾t 𝑆))    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)    &   𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵))
 
8-Jul-2023ennnfonelemh 12359 Lemma for ennnfone 12380. (Contributed by Jim Kingdon, 8-Jul-2023.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ω–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))    &   𝐻 = seq0(𝐺, 𝐽)    ⇒   (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
 
7-Jul-2023seqf2 10420 Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
(𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝐶)
 
3-Jul-2023limcimolemlt 13427 Lemma for limcimo 13428. (Contributed by Jim Kingdon, 3-Jul-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ (𝐾 ↾t 𝑆))    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑 → 𝐷 ∈ ℝ+)    &   (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐵))    &   (𝜑 → 𝑌 ∈ (𝐹 limℂ 𝐵))    &   (𝜑 → ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))    &   (𝜑 → 𝐺 ∈ ℝ+)    &   (𝜑 → ∀𝑤 ∈ 𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))    ⇒   (𝜑 → (abs‘(𝑋 − 𝑌)) < (abs‘(𝑋 − 𝑌)))
 
28-Jun-2023dvfgg 13451 Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and ℂ. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
 
28-Jun-2023dvbsssg 13449 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → dom (𝑆 D 𝐹) ⊆ 𝑆)
 
27-Jun-2023dvbssntrcntop 13447 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ 𝑆)    &   𝐽 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴))
 
27-Jun-2023eldvap 13445 The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
𝑇 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐺 = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 limℂ 𝐵))))
 
27-Jun-2023dvfvalap 13444 Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
𝑇 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))
 
27-Jun-2023dvlemap 13443 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
(𝜑 → 𝐹:𝐷⟶ℂ)    &   (𝜑 → 𝐷 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ((𝜑 ∧ 𝐴 ∈ {𝑤 ∈ 𝐷 ∣ 𝑤 # 𝐵}) → (((𝐹‘𝐴) − (𝐹‘𝐵)) / (𝐴 − 𝐵)) ∈ ℂ)
 
25-Jun-2023reldvg 13442 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → Rel (𝑆 D 𝐹))
 
25-Jun-2023df-dvap 13420 Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of ℂ and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = ℂ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((MetOpen‘(abs ∘ − )) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ {𝑤 ∈ dom 𝑓 ∣ 𝑤 # 𝑥} ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))
 
18-Jun-2023limccnpcntop 13438 If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
(𝜑 → 𝐹:𝐴⟶𝐷)    &   (𝜑 → 𝐷 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐷)    &   (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐵))    &   (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))    ⇒   (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) limℂ 𝐵))
 
18-Jun-2023r19.30dc 2617 Restricted quantifier version of 19.30dc 1620. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
 
17-Jun-2023r19.28v 2598 Restricted quantifier version of one direction of 19.28 1556. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3507.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
 
17-Jun-2023r19.27v 2597 Restricted quantitifer version of one direction of 19.27 1554. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3511.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
 
16-Jun-2023cnlimcim 13434 If 𝐹 is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
(𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) → (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))))
 
16-Jun-2023cncfcn1cntop 13375 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
 
14-Jun-2023cnplimcim 13430 If a function is continuous at 𝐵, its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐴)    ⇒   ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))))
 
14-Jun-2023metcnpd 13314 Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.)
(𝜑 → 𝐽 = (MetOpen‘𝐶))    &   (𝜑 → 𝐾 = (MetOpen‘𝐷))    &   (𝜑 → 𝐶 ∈ (∞Met‘𝑋))    &   (𝜑 → 𝐷 ∈ (∞Met‘𝑌))    &   (𝜑 → 𝑃 ∈ 𝑋)    ⇒   (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
 
6-Jun-2023cntoptop 13327 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   𝐽 ∈ Top
 
6-Jun-2023cntoptopon 13326 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   𝐽 ∈ (TopOn‘ℂ)
 
3-Jun-2023limcdifap 13425 It suffices to consider functions which are not defined at 𝐵 to define the limit of a function. In particular, the value of the original function 𝐹 at 𝐵 does not affect the limit of 𝐹. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    ⇒   (𝜑 → (𝐹 limℂ 𝐵) = ((𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵}) limℂ 𝐵))
 
3-Jun-2023ellimc3ap 13424 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
 
3-Jun-2023df-limced 13419 Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
limℂ = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)))})
 
30-May-2023modprm1div 12201 A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴 mod 𝑃) = 1 ↔ 𝑃 ∥ (𝐴 − 1)))
 
30-May-2023modm1div 11762 An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ) → ((𝐴 mod 𝑁) = 1 ↔ 𝑁 ∥ (𝐴 − 1)))
 
30-May-2023eluz4nn 9527 An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ ℕ)
 
30-May-2023eluz4eluz2 9526 An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘2))
 
29-May-2023mulcncflem 13384 Lemma for mulcncf 13385. (Contributed by Jim Kingdon, 29-May-2023.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    &   (𝜑 → 𝑉 ∈ 𝑋)    &   (𝜑 → 𝐸 ∈ ℝ+)    &   (𝜑 → 𝐹 ∈ ℝ+)    &   (𝜑 → 𝐺 ∈ ℝ+)    &   (𝜑 → 𝑆 ∈ ℝ+)    &   (𝜑 → 𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))    &   (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))    &   (𝜑 → ∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))    ⇒   (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
 
26-May-2023cdivcncfap 13381 Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
𝐹 = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥))    ⇒   (𝐴 ∈ ℂ → 𝐹 ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
 
26-May-2023reccn2ap 11276 The reciprocal function is continuous. The class 𝑇 is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2170. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
𝑇 = (inf({1, ((abs‘𝐴) · 𝐵)}, ℝ, < ) · ((abs‘𝐴) / 2))    ⇒   ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
 
23-May-2023iooretopg 13322 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
 
23-May-2023minclpr 11200 The minimum of two real numbers is one of those numbers if and only if dichotomy (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) holds. For example, this can be combined with zletric 9256 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)))
 
22-May-2023qtopbasss 13315 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
𝑆 ⊆ ℝ*    &   ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → sup({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)    &   ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → inf({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)    ⇒   ((,) “ (𝑆 × 𝑆)) ∈ TopBases
 
22-May-2023iooinsup 11240 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
(((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (sup({𝐴, 𝐶}, ℝ*, < )(,)inf({𝐵, 𝐷}, ℝ*, < )))
 
21-May-2023df-sumdc 11317 Define the sum of a series with an index set of integers 𝐴. The variable 𝑘 is normally a free variable in 𝐵, i.e., 𝐵 can be thought of as 𝐵(𝑘). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an if expression so that we only need 𝐵 to be defined where 𝑘 ∈ 𝐴. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). Examples: Σ𝑘 ∈ {1, 2, 4}𝑘 means 1 + 2 + 4 = 7, and Σ𝑘 ∈ ℕ(1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11485). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
 
19-May-2023bdmopn 13298 The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    &   𝐽 = (MetOpen‘𝐶)    ⇒   ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
 
19-May-2023bdbl 13297 The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
 
19-May-2023bdmet 13296 The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
 
19-May-2023xrminltinf 11235 Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (inf({𝐵, 𝐶}, ℝ*, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴)))
 
19-May-2023clel5 2867 Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
(𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
 
18-May-2023xrminrecl 11236 The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ*, < ) = inf({𝐴, 𝐵}, ℝ, < ))
 
18-May-2023ralnex2 2609 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
 
17-May-2023bdtrilem 11202 Lemma for bdtri 11203. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐵 − 𝐶))) ≤ (𝐶 + (abs‘((𝐴 + 𝐵) − 𝐶))))
 
15-May-2023xrbdtri 11239 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 < 𝐶)) → inf({(𝐴 +𝑒 𝐵), 𝐶}, ℝ*, < ) ≤ (inf({𝐴, 𝐶}, ℝ*, < ) +𝑒 inf({𝐵, 𝐶}, ℝ*, < )))
 
15-May-2023bdtri 11203 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
 
15-May-2023minabs 11199 The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2))
 
11-May-2023xrmaxadd 11224 Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐵, 𝐶}, ℝ*, < )))
 
11-May-2023xrmaxaddlem 11223 Lemma for xrmaxadd 11224. The case where 𝐴 is real. (Contributed by Jim Kingdon, 11-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐵, 𝐶}, ℝ*, < )))
 
10-May-2023xrminadd 11238 Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 inf({𝐵, 𝐶}, ℝ*, < )))
 
10-May-2023xrmaxlesup 11222 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))
 
10-May-2023xrltmaxsup 11220 The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < sup({𝐴, 𝐵}, ℝ*, < ) ↔ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))
 
9-May-2023bdxmet 13295 The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
 
9-May-2023bdmetval 13294 Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ))
 
7-May-2023setsmstsetg 13275 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
(𝜑 → 𝑋 = (Base‘𝑀))    &   (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑 → 𝑀 ∈ 𝑉)    &   (𝜑 → (MetOpen‘𝐷) ∈ 𝑊)    ⇒   (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))
 
6-May-2023dsslid 12578 Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
(dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ)
 
5-May-2023mopnrel 13235 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
Rel MetOpen
 
5-May-2023fsumsersdc 11358 Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))
 
4-May-2023blex 13181 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
(𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V)
 
4-May-2023summodc 11346 A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))    ⇒   (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))
 
4-May-2023summodclem2 11345 Lemma for summodc 11346. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))    ⇒   ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
 
4-May-2023xrminrpcl 11237 The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ+)
 
4-May-2023xrlemininf 11234 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
 
3-May-2023xrltmininf 11233 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
 
3-May-2023xrmineqinf 11232 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → inf({𝐴, 𝐵}, ℝ*, < ) = 𝐵)
 
3-May-2023xrmin2inf 11231 The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐵)
 
3-May-2023xrmin1inf 11230 The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐴)
 
3-May-2023xrmincl 11229 The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ*)
 
2-May-2023xrminmax 11228 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) = -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ))
 
2-May-2023xrnegcon1d 11227 Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
(𝜑 → 𝐴 ∈ ℝ*)    &   (𝜑 → 𝐵 ∈ ℝ*)    ⇒   (𝜑 → (-𝑒𝐴 = 𝐵 ↔ -𝑒𝐵 = 𝐴))
 
2-May-2023infxrnegsupex 11226 The infimum of a set of extended reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
(𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
 
2-May-2023xrnegiso 11225 Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
𝐹 = (𝑥 ∈ ℝ* ↦ -𝑒𝑥)    ⇒   (𝐹 Isom < , ◡ < (ℝ*, ℝ*) ∧ ◡𝐹 = 𝐹)
 
30-Apr-2023xrmaxltsup 11221 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))
 
30-Apr-2023xrmaxrecl 11218 The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ*, < ) = sup({𝐴, 𝐵}, ℝ, < ))
 
30-Apr-2023xrmax2sup 11217 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ*, < ))
 
30-Apr-2023xrmax1sup 11216 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ*, < ))
 
29-Apr-2023xrmaxcl 11215 The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → sup({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ*)
 
29-Apr-2023xrmaxiflemval 11213 Lemma for xrmaxif 11214. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
𝑀 = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))    ⇒   ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑀 ∈ ℝ* ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑀 < 𝑥 ∧ ∀𝑥 ∈ ℝ* (𝑥 < 𝑀 → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧)))
 
29-Apr-2023xrmaxiflemcom 11212 Lemma for xrmaxif 11214. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
 
29-Apr-2023xrmaxiflemcl 11208 Lemma for xrmaxif 11214. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*)
 
29-Apr-2023sbco2v 1941 Version of sbco2 1958 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
Ⅎ𝑧𝜑    ⇒   ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
28-Apr-2023xrmaxiflemlub 11211 Lemma for xrmaxif 11214. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 28-Apr-2023.)
(𝜑 → 𝐴 ∈ ℝ*)    &   (𝜑 → 𝐵 ∈ ℝ*)    &   (𝜑 → 𝐶 ∈ ℝ*)    &   (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))    ⇒   (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
 
26-Apr-2023xrmaxif 11214 Maximum of two extended reals in terms of if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → sup({𝐴, 𝐵}, ℝ*, < ) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
 
26-Apr-2023xrmaxiflemab 11210 Lemma for xrmaxif 11214. A variation of xrmaxleim 11207- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
(𝜑 → 𝐴 ∈ ℝ*)    &   (𝜑 → 𝐵 ∈ ℝ*)    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
 
26-Apr-2023xrmaxifle 11209 An upper bound for {𝐴, 𝐵} in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
 
25-Apr-2023xrmaxleim 11207 Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ*, < ) = 𝐵))
 
25-Apr-2023rpmincl 11201 The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)
 
25-Apr-2023mincl 11194 The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℝ)
 
24-Apr-2023psmetrel 13116 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
Rel PsMet
 
23-Apr-2023bcval5 10697 Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁 − 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁 − 𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
 
23-Apr-2023ser3le 10474 Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))
 
23-Apr-2023seq3z 10467 If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍)    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → (𝐹‘𝐾) = 𝑍)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
 
23-Apr-2023seq3caopr 10439 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
 
23-Apr-2023seq3caopr2 10438 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
 
22-Apr-2023ser3sub 10462 The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁)))
 
22-Apr-2023seq3caopr3 10437 Lemma for seq3caopr2 10438. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    &   ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
 
22-Apr-2023ser3mono 10434 The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥))    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))
 
21-Apr-2023metrtri 13171 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
 
21-Apr-2023sqxpeq0 5034 A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)
((𝐴 × 𝐴) = ∅ ↔ 𝐴 = ∅)
 
20-Apr-2023xmetrel 13137 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel ∞Met
 
20-Apr-2023metrel 13136 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel Met
 
19-Apr-2023psmetge0 13125 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
 
18-Apr-2023xleaddadd 9844 Cancelling a factor of two in ≤ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐴) ≤ (𝐵 +𝑒 𝐵)))
 
17-Apr-2023xposdif 9839 Extended real version of posdif 8374. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
 
17-Apr-2023nmnfgt 9775 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* → (-∞ < 𝐴 ↔ 𝐴 ≠ -∞))
 
17-Apr-2023npnflt 9772 An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞))
 
16-Apr-2023xltadd1 9833 Extended real version of ltadd1 8348. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
 
13-Apr-2023xrmnfdc 9800 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
 
13-Apr-2023xrpnfdc 9799 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
 
11-Apr-2023dmxpid 4832 The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
dom (𝐴 × 𝐴) = 𝐴
 
9-Apr-2023isumz 11352 Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
(((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0)
 
9-Apr-2023summodclem2a 11344 Lemma for summodc 11346. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))    &   𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))    ⇒   (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))
 
9-Apr-2023summodclem3 11343 Lemma for summodc 11346. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))    &   𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))    ⇒   (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
 
9-Apr-2023sumrbdc 11342 Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → DECID 𝑘 ∈ 𝐴)    ⇒   (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶))
 
9-Apr-2023seq3coll 10777 The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑 → 𝑍 ∈ 𝑆)    &   (𝜑 → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))    &   (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘1)) → (𝐻‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)    &   ((𝜑 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘𝑁)) = (seq1( + , 𝐻)‘𝑁))
 
8-Apr-2023zsumdc 11347 Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ⊆ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   (𝜑 → ∀𝑥 ∈ 𝑍 DECID 𝑥 ∈ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
 
8-Apr-2023sumrbdclem 11340 Lemma for sumrbdc 11342. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹))
 
8-Apr-2023isermulc2 11303 Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘)))    ⇒   (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴))
 
8-Apr-2023seq3id 10464 Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥)    &   (𝜑 → 𝑍 ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → (𝐹‘𝑁) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹))
 
8-Apr-2023seq3id3 10463 A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)    &   (𝜑 → 𝑍 ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
 
7-Apr-2023seq3shft2 10429 Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐾 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
 
7-Apr-2023seq3feq 10428 Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
 
7-Apr-2023r19.2m 3501 Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1631). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
 
6-Apr-2023lmtopcnp 13044 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
(𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   (𝜑 → 𝐾 ∈ Top)    &   (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))    ⇒   (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))
 
6-Apr-2023cnptoprest2 13034 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
𝑋 = ∪ 𝐽    &   𝑌 = ∪ 𝐾    ⇒   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))
 
5-Apr-2023cnptoprest 13033 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
𝑋 = ∪ 𝐽    &   𝑌 = ∪ 𝐾    ⇒   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)))
 
4-Apr-2023exmidmp 7133 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
(EXMID → ω ∈ Markov)
 
2-Apr-2023sup3exmid 8873 If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ 𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧)))    ⇒   DECID 𝜑
 
31-Mar-2023cnptopresti 13032 One direction of cnptoprest 13033 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))
 
30-Mar-2023cncnp2m 13025 A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
𝑋 = ∪ 𝐽    &   𝑌 = ∪ 𝐾    ⇒   ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
 
29-Mar-2023exmidlpo 7119 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
(EXMID → ω ∈ Omni)
 
28-Mar-2023icnpimaex 13005 Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))
 
28-Mar-2023cnpf2 13001 A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
 
28-Mar-2023cnprcl2k 13000 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
 
27-Mar-2023mptrcl 5578 Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)
 
25-Mar-2023lmreltop 12987 The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝐽 ∈ Top → Rel (⇝𝑡‘𝐽))
 
25-Mar-2023fodjumkv 7136 A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝜑 → 𝑀 ∈ Markov)    &   (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))    ⇒   (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))
 
25-Mar-2023fodjumkvlemres 7135 Lemma for fodjumkv 7136. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝜑 → 𝑀 ∈ Markov)    &   (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))    &   𝑃 = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    ⇒   (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))
 
25-Mar-2023fodju0 7123 Lemma for fodjuomni 7125 and fodjumkv 7136. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)    ⇒   (𝜑 → 𝐴 = ∅)
 
25-Mar-2023fodjum 7122 Lemma for fodjuomni 7125 and fodjumkv 7136. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   (𝜑 → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)    ⇒   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
 
25-Mar-2023fodjuf 7121 Lemma for fodjuomni 7125 and fodjumkv 7136. Domain and range of 𝑃. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   (𝜑 → 𝑂 ∈ 𝑉)    ⇒   (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
 
23-Mar-2023restrcl 12961 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
 
22-Mar-2023neipsm 12948 A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
𝑋 = ∪ 𝐽    ⇒   ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ∃𝑥 𝑥 ∈ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
 
19-Mar-2023mkvprop 7134 Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = ω case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
((𝐴 ∈ Markov ∧ ∀𝑛 ∈ 𝐴 DECID 𝜑 ∧ ¬ ∀𝑛 ∈ 𝐴 ¬ 𝜑) → ∃𝑛 ∈ 𝐴 𝜑)
 
18-Mar-2023omnimkv 7132 An omniscient set is Markov. In particular, the case where 𝐴 is ω means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ Omni → 𝐴 ∈ Markov)
 
18-Mar-2023ismkvmap 7130 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
 
18-Mar-2023ismkv 7129 The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
 
18-Mar-2023df-markov 7128 A Markov set is one where if a predicate (here represented by a function 𝑓) on that set does not hold (where hold means is equal to 1o) for all elements, then there exists an element where it fails (is equal to ∅). Generalization of definition 2.5 of [Pierik], p. 9.

In particular, ω ∈ Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)

Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))}
 
17-Mar-2023finct 7093 A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
(𝐴 ∈ Fin → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
 
16-Mar-2023ctmlemr 7085 Lemma for ctm 7086. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
(∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
 
15-Mar-2023caseinl 7068 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
(𝜑 → 𝐹 Fn 𝐵)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))
 
13-Mar-2023enumct 7092 A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as ∃𝑔𝑔:ω–onto→(𝐴 ⊔ 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
 
13-Mar-2023enumctlemm 7091 Lemma for enumct 7092. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
(𝜑 → 𝐹:𝑁–onto→𝐴)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → ∅ ∈ 𝑁)    &   𝐺 = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)))    ⇒   (𝜑 → 𝐺:ω–onto→𝐴)
 
13-Mar-2023ctm 7086 Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→𝐴))
 
13-Mar-20230ct 7084 The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o)
 
13-Mar-2023ctex 6731 A class dominated by ω is a set. See also ctfoex 7095 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
(𝐴 ≼ ω → 𝐴 ∈ V)
 
12-Mar-2023cls0 12927 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
(𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
 
12-Mar-2023algrp1 12000 The value of the algorithm iterator 𝑅 at (𝐾 + 1). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
𝑍 = (ℤ≥‘𝑀)    &   𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆⟶𝑆)    ⇒   ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾)))
 
12-Mar-2023ialgr0 11998 The value of the algorithm iterator 𝑅 at 0 is the initial state 𝐴. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
𝑍 = (ℤ≥‘𝑀)    &   𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆⟶𝑆)    ⇒   (𝜑 → (𝑅‘𝑀) = 𝐴)
 
11-Mar-2023ntreq0 12926 Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = ∪ 𝐽    ⇒   ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))
 
11-Mar-2023clstop 12921 The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
𝑋 = ∪ 𝐽    ⇒   (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
 
11-Mar-2023ntrss 12913 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = ∪ 𝐽    ⇒   ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
 
10-Mar-2023iuncld 12909 A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
𝑋 = ∪ 𝐽    ⇒   ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
 
5-Mar-20232basgeng 12876 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
((𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
 
5-Mar-2023exmidsssn 4188 Excluded middle is equivalent to the biconditionalized version of sssnr 3740 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
 
5-Mar-2023exmidn0m 4187 Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
 
4-Mar-2023eltg3 12851 Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥)))
 
4-Mar-2023tgvalex 12844 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
(𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
 
4-Mar-2023biadanii 608 Inference associated with biadani 607. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
(𝜑 → 𝜓)    &   (𝜓 → (𝜑 ↔ 𝜒))    ⇒   (𝜑 ↔ (𝜓 ∧ 𝜒))
 
4-Mar-2023biadani 607 An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
(𝜑 → 𝜓)    ⇒   ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
 
16-Feb-2023ixp0 6709 The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
(∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)
 
16-Feb-2023ixpm 6708 If an infinite Cartesian product of a family 𝐵(𝑥) is inhabited, every 𝐵(𝑥) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
(∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
 
16-Feb-2023exmidundifim 4193 Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4192 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
(EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
 
15-Feb-2023ixpintm 6703 The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦)
 
15-Feb-2023ixpiinm 6702 The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)
 
15-Feb-2023ixpexgg 6700 The existence of an infinite Cartesian product. 𝑥 is normally a free-variable parameter in 𝐵. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
((𝐴 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 ∈ V)
 
15-Feb-2023nfixpxy 6695 Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    ⇒   Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵
 
13-Feb-2023topnpropgd 12593 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    &   (𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐿 ∈ 𝑊)    ⇒   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
 
12-Feb-2023slotex 12443 Existence of slot value. A corollary of slotslfn 12442. (Contributed by Jim Kingdon, 12-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    ⇒   (𝐴 ∈ 𝑉 → (𝐸‘𝐴) ∈ V)
 
11-Feb-2023topnvalg 12591 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)    ⇒   (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
 
10-Feb-2023slotslfn 12442 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    ⇒   𝐸 Fn V
 
9-Feb-2023pleslid 12575 Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
(le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
 
9-Feb-2023topgrptsetd 12572 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → 𝐽 ∈ 𝑋)    ⇒   (𝜑 → 𝐽 = (TopSet‘𝑊))
 
9-Feb-2023topgrpplusgd 12571 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → 𝐽 ∈ 𝑋)    ⇒   (𝜑 → + = (+g‘𝑊))
 
9-Feb-2023topgrpbasd 12570 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → 𝐽 ∈ 𝑋)    ⇒   (𝜑 → 𝐵 = (Base‘𝑊))
 
9-Feb-2023topgrpstrd 12569 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → 𝐽 ∈ 𝑋)    ⇒   (𝜑 → 𝑊 Struct ⟨1, 9⟩)
 
9-Feb-2023tsetslid 12568 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
(TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
 
8-Feb-2023ipsipd 12565 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → 𝐼 = (·𝑖‘𝐴))
 
8-Feb-2023ipsvscad 12564 The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → · = ( ·𝑠 ‘𝐴))
 
8-Feb-2023ipsscad 12563 The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → 𝑆 = (Scalar‘𝐴))
 
7-Feb-2023ipsmulrd 12562 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → × = (.r‘𝐴))
 
7-Feb-2023ipsaddgd 12561 The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → + = (+g‘𝐴))
 
7-Feb-2023ipsbased 12560 The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → 𝐵 = (Base‘𝐴))
 
7-Feb-2023ipsstrd 12559 A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → × ∈ 𝑋)    &   (𝜑 → 𝑆 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑄)    &   (𝜑 → 𝐼 ∈ 𝑍)    ⇒   (𝜑 → 𝐴 Struct ⟨1, 8⟩)
 
7-Feb-2023ipslid 12558 Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
(·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
 
7-Feb-2023lmodvscad 12555 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑍)    ⇒   (𝜑 → · = ( ·𝑠 ‘𝑊))
 
6-Feb-2023lmodscad 12554 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑍)    ⇒   (𝜑 → 𝐹 = (Scalar‘𝑊))
 
6-Feb-2023lmodplusgd 12553 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑍)    ⇒   (𝜑 → + = (+g‘𝑊))
 
6-Feb-2023lmodbased 12552 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑍)    ⇒   (𝜑 → 𝐵 = (Base‘𝑊))
 
5-Feb-2023lmodstrd 12551 A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑌)    &   (𝜑 → · ∈ 𝑍)    ⇒   (𝜑 → 𝑊 Struct ⟨1, 6⟩)
 
5-Feb-2023vscaslid 12550 Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
 
5-Feb-2023scaslid 12547 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
(Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
 
5-Feb-2023srngplusgd 12542 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → · ∈ 𝑋)    &   (𝜑 → ∗ ∈ 𝑌)    ⇒   (𝜑 → + = (+g‘𝑅))
 
5-Feb-2023srngbased 12541 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → · ∈ 𝑋)    &   (𝜑 → ∗ ∈ 𝑌)    ⇒   (𝜑 → 𝐵 = (Base‘𝑅))
 
5-Feb-2023srngstrd 12540 A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → + ∈ 𝑊)    &   (𝜑 → · ∈ 𝑋)    &   (𝜑 → ∗ ∈ 𝑌)    ⇒   (𝜑 → 𝑅 Struct ⟨1, 4⟩)
 
5-Feb-2023opelstrsl 12514 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑 → 𝑆 Struct 𝑋)    &   (𝜑 → 𝑉 ∈ 𝑌)    &   (𝜑 → ⟨(𝐸‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   (𝜑 → 𝑉 = (𝐸‘𝑆))
 
4-Feb-2023starvslid 12539 Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
(*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)
 
3-Feb-2023rngbaseg 12534 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅))
 
3-Feb-2023rngstrg 12533 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct ⟨1, 3⟩)
 
3-Feb-2023mulrslid 12530 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
(.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
 
3-Feb-2023plusgslid 12513 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
 
2-Feb-20232strop1g 12523 The other slot of a constructed two-slot structure. Version of 2stropg 12520 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    &   𝐸 = Slot 𝑁    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
 
2-Feb-20232strbas1g 12522 The base set of a constructed two-slot structure. Version of 2strbasg 12519 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
 
2-Feb-20232strstr1g 12521 A constructed two-slot structure. Version of 2strstrg 12518 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
 
31-Jan-2023baseslid 12472 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
 
31-Jan-2023strsl0 12464 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    ⇒   ∅ = (𝐸‘∅)
 
31-Jan-2023strslss 12463 Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
𝑇 ∈ V    &   Fun 𝑇    &   𝑆 ⊆ 𝑇    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆    ⇒   (𝐸‘𝑇) = (𝐸‘𝑆)
 
31-Jan-2023strslssd 12462 Deduction version of strslss 12463. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑 → 𝑇 ∈ 𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑 → 𝑆 ⊆ 𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆))
 
30-Jan-2023strslfv3 12461 Variant on strslfv 12460 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝜑 → 𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑 → 𝐶 ∈ 𝑉)    &   𝐴 = (𝐸‘𝑈)    ⇒   (𝜑 → 𝐴 = 𝐶)
 
30-Jan-2023strslfv 12460 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12422). By virtue of ndxslid 12441, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 Struct 𝑋    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    ⇒   (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))
 
30-Jan-2023strslfv2 12459 A variation on strslfv 12460 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 ∈ V    &   Fun ◡◡𝑆    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆    ⇒   (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))
 
30-Jan-2023strslfv2d 12458 Deduction version of strslfv 12460. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun ◡◡𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑊)    ⇒   (𝜑 → 𝐶 = (𝐸‘𝑆))
 
30-Jan-2023strslfvd 12457 Deduction version of strslfv 12460. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   (𝜑 → 𝐶 = (𝐸‘𝑆))
 
30-Jan-2023strsetsid 12449 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑 → 𝑆 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)    ⇒   (𝜑 → 𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸‘𝑆)⟩))
 
30-Jan-2023funresdfunsndc 6485 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
((∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)
 
29-Jan-2023ndxslid 12441 A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12460. (Contributed by Jim Kingdon, 29-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    ⇒   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
 
29-Jan-2023fnsnsplitdc 6484 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
 
28-Jan-20232stropg 12520 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
 
28-Jan-20232strbasg 12519 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
 
28-Jan-20232strstrg 12518 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨1, 𝑁⟩)
 
28-Jan-20231strstrg 12516 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐺 Struct ⟨1, 1⟩)
 
27-Jan-2023strle2g 12509 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐵 = 𝐽    ⇒   ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩)
 
27-Jan-2023strle1g 12508 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼    ⇒   (𝑋 ∈ 𝑉 → {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩)
 
27-Jan-2023strleund 12506 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
(𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   (𝜑 → 𝐺 Struct ⟨𝐶, 𝐷⟩)    &   (𝜑 → 𝐵 < 𝐶)    ⇒   (𝜑 → (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩)
 
26-Jan-2023ressid2 12477 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐵 = (Base‘𝑊)    ⇒   ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)
 
24-Jan-2023setsslnid 12467 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ 𝐷    &   𝐷 ∈ ℕ    ⇒   ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))
 
24-Jan-2023setsslid 12466 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    ⇒   ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
 
22-Jan-2023setsabsd 12455 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑊)    &   (𝜑 → 𝐵 ∈ 𝑋)    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
 
22-Jan-2023setsresg 12454 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
 
22-Jan-2023setsex 12448 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
 
22-Jan-20232zsupmax 11189 Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
 
22-Jan-2023elpwpwel 4460 A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
(𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)
 
21-Jan-2023funresdfunsnss 5699 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) ⊆ 𝐹)
 
20-Jan-2023setsvala 12447 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
 
20-Jan-2023fnsnsplitss 5695 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) ⊆ 𝐹)
 
19-Jan-2023strfvssn 12438 A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆)
 
19-Jan-2023strnfvn 12437 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12422) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12460. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    ⇒   (𝐸‘𝑆) = (𝑆‘𝑁)
 
19-Jan-2023strnfvnd 12436 Deduction version of strnfvn 12437. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
 
18-Jan-2023isstructr 12431 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
 
18-Jan-2023isstructim 12430 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct ⟨𝑀, 𝑁⟩ → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
 
18-Jan-2023isstruct2r 12427 The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)
 
18-Jan-2023isstruct2im 12426 The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
 
18-Jan-2023sbiev 1785 Conversion of implicit substitution to explicit substitution. Version of sbie 1784 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
 
16-Jan-2023toponsspwpwg 12814 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
 
14-Jan-2023istopfin 12792 Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12791. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽)))
 
14-Jan-2023fiintim 6906 If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as 𝑥 and 𝑦 not being equal, or 𝐴 having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐴))
 
9-Jan-2023divccncfap 13371 Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))    ⇒   ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐹 ∈ (ℂ–cn→ℂ))
 
7-Jan-2023eap1 11748 e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 1
 
7-Jan-2023eap0 11746 e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 0
 
7-Jan-2023egt2lt3 11742 Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
(2 < e ∧ e < 3)
 
6-Jan-2023eirr 11741 e is not rational. In the absence of excluded middle, we can distinguish between this and saying that e is irrational in the sense of being apart from any rational number, which is eirrap 11740. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
e ∉ ℚ
 
6-Jan-2023eirrap 11740 e is irrational. That is, for any rational number, e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that e is not rational, which is eirr 11741. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝑄 ∈ ℚ → e # 𝑄)
 
6-Jan-2023btwnapz 9342 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℤ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐵 < (𝐴 + 1))    ⇒   (𝜑 → 𝐵 # 𝐶)
 
6-Jan-2023apmul2 8706 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐶 · 𝐴) # (𝐶 · 𝐵)))
 
1-Jan-2023nnap0i 8909 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
𝐴 ∈ ℕ    ⇒   𝐴 # 0
 
31-Dec-20222logb9irrALT 13686 Alternate proof of 2logb9irr 13683: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(2 logb 9) ∈ (ℝ ∖ ℚ)
 
31-Dec-20222logb3irr 13685 Example for logbprmirr 13684. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
(2 logb 3) ∈ (ℝ ∖ ℚ)
 
31-Dec-2022logbprmirr 13684 The logarithm of a prime to a different prime base is not rational. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 13685). (Contributed by AV, 31-Dec-2022.)
((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋 ≠ 𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))
 
30-Dec-2022elpqb 9608 A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
 
29-Dec-2022sqrt2cxp2logb9e3 13687 The square root of two to the power of the logarithm of nine to base two is three. (√‘2) and (2 logb 9) are not rational (see sqrt2irr0 12118 resp. 2logb9irr 13683), satisfying the statement in 2irrexpq 13688. (Contributed by AV, 29-Dec-2022.)
((√‘2)↑𝑐(2 logb 9)) = 3
 
29-Dec-20222logb9irr 13683 Example for logbgcd1irr 13679. The logarithm of nine to base two is not rational. Also see 2logb9irrap 13689 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
(2 logb 9) ∈ (ℝ ∖ ℚ)
 
29-Dec-2022logbgcd1irrap 13682 The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, (2 logb 9) # 𝑄 where 𝑄 is rational. (Contributed by AV, 29-Dec-2022.)
(((𝑋 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) ∧ ((𝑋 gcd 𝐵) = 1 ∧ 𝑄 ∈ ℚ)) → (𝐵 logb 𝑋) # 𝑄)
 
29-Dec-2022logbgcd1irr 13679 The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, (2 logb 9) ∈ (ℝ ∖ ℚ). (Contributed by AV, 29-Dec-2022.)
((𝑋 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))
 
29-Dec-2022logbgt0b 13678 The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ (𝐵 ∈ ℝ+ ∧ 1 < 𝐵)) → (0 < (𝐵 logb 𝐴) ↔ 1 < 𝐴))
 
29-Dec-2022cxpcom 13651 Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴↑𝑐𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶)↑𝑐𝐵))
 
29-Dec-2022elpq 9607 A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
 
26-Dec-2022apdivmuld 8730 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴))
 
25-Dec-2022tanaddaplem 11701 A useful intermediate step in tanaddap 11702 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((cos‘𝐴) # 0 ∧ (cos‘𝐵) # 0)) → ((cos‘(𝐴 + 𝐵)) # 0 ↔ ((tan‘𝐴) · (tan‘𝐵)) # 1))
 
25-Dec-2022subap0 8562 Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) # 0 ↔ 𝐴 # 𝐵))
 
23-Dec-20222irrexpq 13688 There exist real numbers 𝑎 and 𝑏 which are not rational such that (𝑎↑𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named non-rational numbers (√‘2) and (2 logb 9), see sqrt2irr0 12118, 2logb9irr 13683 and sqrt2cxp2logb9e3 13687. Therefore, this proof is acceptable/usable in intuitionistic logic.

For a theorem which is the same but proves that 𝑎 and 𝑏 are irrational (in the sense of being apart from any rational number), see 2irrexpqap 13690. (Contributed by AV, 23-Dec-2022.)

∃𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎↑𝑐𝑏) ∈ ℚ
 
23-Dec-2022rpcxpsqrtth 13644 Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 10995. (Contributed by AV, 23-Dec-2022.)
(𝐴 ∈ ℝ+ → ((√‘𝐴)↑𝑐2) = 𝐴)
 
23-Dec-2022sqrt2irr0 12118 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
(√‘2) ∈ (ℝ ∖ ℚ)
 
22-Dec-2022tanval3ap 11677 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ ℂ ∧ ((exp‘(2 · (i · 𝐴))) + 1) # 0) → (tan‘𝐴) = (((exp‘(2 · (i · 𝐴))) − 1) / (i · ((exp‘(2 · (i · 𝐴))) + 1))))
 
22-Dec-2022tanval2ap 11676 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (i · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
 
22-Dec-2022tanclapd 11675 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (cos‘𝐴) # 0)    ⇒   (𝜑 → (tan‘𝐴) ∈ ℂ)
 
21-Dec-2022tanclap 11672 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) ∈ ℂ)
 
21-Dec-2022tanvalap 11671 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
 
20-Dec-2022reef11 11662 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) = (exp‘𝐵) ↔ 𝐴 = 𝐵))
 
20-Dec-2022efltim 11661 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (exp‘𝐴) < (exp‘𝐵)))
 
20-Dec-2022eqord1 8402 A strictly increasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))    ⇒   ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁))
 
14-Dec-2022iserabs 11438 Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))    ⇒   (𝜑 → (abs‘𝐴) ≤ 𝐵)
 
12-Dec-2022efap0 11640 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
(𝐴 ∈ ℂ → (exp‘𝐴) # 0)
 
8-Dec-2022efcllem 11622 Lemma for efcl 11627. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
 
8-Dec-2022efcllemp 11621 Lemma for efcl 11627. The series that defines the exponential function converges. The ratio test cvgratgt0 11496 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → (2 · (abs‘𝐴)) < 𝐾)    ⇒   (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
 
8-Dec-2022eftvalcn 11620 The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))
 
8-Dec-2022mertensabs 11500 Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then (Σ𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)
((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)    &   ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))    &   ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))    &   (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )    ⇒   (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
 
3-Dec-2022mertenslemub 11497 Lemma for mertensabs 11500. An upper bound for 𝑇. (Contributed by Jim Kingdon, 3-Dec-2022.)
((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}    &   (𝜑 → 𝑋 ∈ 𝑇)    &   (𝜑 → 𝑆 ∈ ℕ)    ⇒   (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
 
2-Dec-2022mertenslemi1 11498 Lemma for mertensabs 11500. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)    &   ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))    &   ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))    &   (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   (𝜑 → 𝐸 ∈ ℝ+)    &   𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}    &   (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))    &   (𝜑 → 𝑃 ∈ ℝ)    &   (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (𝑃 + 1)))))    &   (𝜑 → 0 ≤ 𝑃)    &   (𝜑 → ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑃)    ⇒   (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
 
2-Dec-2022fsum3cvg3 11359 A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐴 ⊆ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
2-Dec-2022fsum3cvg2 11357 The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
 
24-Nov-2022cvgratnnlembern 11486 Lemma for cvgratnn 11494. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 𝑀 ∈ ℕ)    ⇒   (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
 
23-Nov-2022cvgratnnlemfm 11492 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 23-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    ⇒   (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))
 
23-Nov-2022cvgratnnlemsumlt 11491 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 23-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
 
21-Nov-2022cvgratnnlemrate 11493 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 21-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀))
 
21-Nov-2022cvgratnnlemabsle 11490 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 21-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
 
21-Nov-2022cvgratnnlemseq 11489 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 21-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
 
15-Nov-2022cvgratnnlemmn 11488 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 15-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))
 
15-Nov-2022cvgratnnlemnexp 11487 Lemma for cvgratnn 11494. (Contributed by Jim Kingdon, 15-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
 
12-Nov-2022cvgratnn 11494 Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms, then the infinite sum of the terms of 𝐹 converges to a complex number. Although this theorem is similar to cvgratz 11495 and cvgratgt0 11496, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11313 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
 
12-Nov-2022fsum3cvg 11341 The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
 
12-Nov-2022seq3id2 10465 The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)    &   (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
 
11-Nov-2022cvgratgt0 11496 Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms beyond some index 𝐵, then the infinite sum of the terms of 𝐹 converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
𝑍 = (ℤ≥‘𝑀)    &   𝑊 = (ℤ≥‘𝑁)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 𝑁 ∈ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
11-Nov-2022cvgratz 11495 Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms, then the infinite sum of the terms of 𝐹 converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 < 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
4-Nov-2022seq3val 10414 Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10417, seq3-1 10416 and seq3p1 10418, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
(𝜑 → 𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
 
4-Nov-2022df-seqfrec 10402 Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as ℕ or ℕ0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seqf 10417, seq3-1 10416 and seq3p1 10418. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series. seq𝑀( + , 𝐹) ⇝ 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 11256), by climdm 11258 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ⇝ ‘seq1( + , 𝐹)) (provided the sequence converges) and evaluates to 2 in this example.

Internally, the frec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
 
3-Nov-2022seq3f1o 10460 Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
3-Nov-2022seq3m1 10424 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)))
 
29-Oct-2022absgtap 11473 Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℝ+)    &   (𝜑 → 𝐵 < (abs‘𝐴))    ⇒   (𝜑 → 𝐴 # 𝐵)
 
29-Oct-2022absltap 11472 Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) < 𝐵)    ⇒   (𝜑 → 𝐴 # 𝐵)
 
29-Oct-20221ap2 9085 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
1 # 2
 
28-Oct-2022expcnv 11467 A sequence of powers of a complex number 𝐴 with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) < 1)    ⇒   (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
 
28-Oct-2022expcnvre 11466 A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 1)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
 
27-Oct-2022ennnfone 12380 A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that 𝐴 is countable (that's the 𝑓:ℕ0–onto→𝐴 part, as seen in theorems like ctm 7086), infinite (that's the part about being able to find an element of 𝐴 distinct from any mapping of a natural number via 𝑓), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
(𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))
 
27-Oct-2022ennnfonelemim 12379 Lemma for ennnfone 12380. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)
(𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))
 
27-Oct-2022ennnfonelemr 12378 Lemma for ennnfone 12380. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ℕ0–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))    ⇒   (𝜑 → 𝐴 ≈ ℕ)
 
27-Oct-2022ennnfonelemnn0 12377 Lemma for ennnfone 12380. A version of ennnfonelemen 12376 expressed in terms of ℕ0 instead of ω. (Contributed by Jim Kingdon, 27-Oct-2022.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:ℕ0–onto→𝐴)    &   (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))    &   𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝜑 → 𝐴 ≈ ℕ)
 
24-Oct-2022pwm1geoserap1 11471 The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)))
 
24-Oct-2022geoserap 11470 The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 1)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)))
 
24-Oct-2022geosergap 11469 The value of the finite geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1) +... + 𝐴↑(𝑁 − 1). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 1)    &   (𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴)))
 
23-Oct-2022expcnvap0 11465 A sequence of powers of a complex number 𝐴 with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) < 1)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
 
22-Oct-2022divcnv 11460 The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
(𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0)
 
22-Oct-2022impcomd 253 Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.)
(𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃))
 
21-Oct-2022isumsplit 11454 Split off the first 𝑁 terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
𝑍 = (ℤ≥‘𝑀)    &   𝑊 = (ℤ≥‘𝑁)    &   (𝜑 → 𝑁 ∈ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ 𝑊 𝐴))
 
21-Oct-2022seq3split 10435 Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
 
20-Oct-2022fidcenumlemrk 6931 Lemma for fidcenum 6933. (Contributed by Jim Kingdon, 20-Oct-2022.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:𝑁–onto→𝐴)    &   (𝜑 → 𝐾 ∈ ω)    &   (𝜑 → 𝐾 ⊆ 𝑁)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
 
20-Oct-2022fidcenumlemrks 6930 Lemma for fidcenum 6933. Induction step for fidcenumlemrk 6931. (Contributed by Jim Kingdon, 20-Oct-2022.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:𝑁–onto→𝐴)    &   (𝜑 → 𝐽 ∈ ω)    &   (𝜑 → suc 𝐽 ⊆ 𝑁)    &   (𝜑 → (𝑋 ∈ (𝐹 “ 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐽)))    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
 
19-Oct-2022fidcenum 6933 A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝐴 ∈ Fin ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))
 
19-Oct-2022fidcenumlemr 6932 Lemma for fidcenum 6933. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝐹:𝑁–onto→𝐴)    &   (𝜑 → 𝑁 ∈ ω)    ⇒   (𝜑 → 𝐴 ∈ Fin)
 
19-Oct-2022fidcenumlemim 6929 Lemma for fidcenum 6933. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))
 
17-Oct-2022iser3shft 11309 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴))
 
17-Oct-2022seq3shft 10802 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁))) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁))
 
16-Oct-2022resqrexlemf1 10972 Lemma for resqrex 10990. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → (𝐹‘1) = (1 + 𝐴))
 
16-Oct-2022resqrexlemf 10971 Lemma for resqrex 10990. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → 𝐹:ℕ⟶ℝ+)
 
16-Oct-2022resqrexlemp1rp 10970 Lemma for resqrex 10990. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10417 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ (𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵(𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝐶) ∈ ℝ+)
 
16-Oct-2022resqrexlem1arp 10969 Lemma for resqrex 10990. 1 + 𝐴 is a positive real (expressed in a way that will help apply seqf 10417 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+)
 
15-Oct-2022inffz 14101 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
(𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
 
15-Oct-2022supfz 14100 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
(𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)
 
12-Oct-2022fsumlessfi 11423 A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.)
(𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   (𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐶 ∈ Fin)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)
 
12-Oct-2022modfsummodlemstep 11420 Induction step for modfsummod 11421. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ)    &   (𝜑 → ¬ 𝑧 ∈ 𝐴)    &   (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁))    ⇒   (𝜑 → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁))
 
10-Oct-2022fsum3 11350 The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀))
 
10-Oct-2022fsumgcl 11349 Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)    ⇒   (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)
 
10-Oct-2022seq3distr 10469 The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
 
10-Oct-2022seq3homo 10466 Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    ⇒   (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
 
8-Oct-2022fsum2dlemstep 11397 Lemma for fsum2d 11398- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦 ∈ 𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜑 → 𝑥 ∈ Fin)    &   (𝜓 ↔ Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)    ⇒   ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 
7-Oct-2022iunfidisj 6923 The finite union of disjoint finite sets is finite. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
 
7-Oct-2022disjnims 3981 If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
(Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
 
6-Oct-2022disjnim 3980 If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
(𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   (Disj 𝑖 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
 
5-Oct-2022dcun 3525 The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
(𝜑 → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → DECID 𝑘 ∈ 𝐵)    ⇒   (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
 
4-Oct-2022ser3add 10461 The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
 
3-Oct-2022seq3-1 10416 Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
(𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
 
3-Oct-2022brrelex12i 4653 Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)
Rel 𝑅    ⇒   (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
1-Oct-2022fsum3ser 11360 A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11375 and fsump1 11383, which should make our notation clear and from which, along with closure fsumcl 11363, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = 𝐴)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ)    ⇒   (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))
 
1-Oct-2022tpfidisj 6905 A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐶)    &   (𝜑 → 𝐵 ≠ 𝐶)    ⇒   (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin)
 
30-Sep-2022exdistrv 1903 Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1895 and 19.42v 1899. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1925. (Contributed by BJ, 30-Sep-2022.)
(∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
28-Sep-2022seq3clss 10423 Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑 → 𝑆 ⊆ 𝑇)    &   ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)
 
27-Sep-2022zmaxcl 11188 The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℤ)
 
24-Sep-2022isumss2 11356 Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set 𝐴 and the added zeroes compose the rest of the containing set 𝐵 which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)    &   (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)    &   (𝜑 → ((𝑀 ∈ ℤ ∧ 𝐵 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵) ∨ 𝐵 ∈ Fin))    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 𝐶, 0))
 
24-Sep-2022preimaf1ofi 6928 The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
(𝜑 → 𝐶 ⊆ 𝐵)    &   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    &   (𝜑 → 𝐶 ∈ Fin)    ⇒   (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin)
 
24-Sep-2022ifmdc 3565 If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
(𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)
 
24-Sep-2022bianassc 467 An inference to merge two lists of conjuncts. (Contributed by Peter Mazsa, 24-Sep-2022.)
(𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ((𝜂 ∧ 𝜑) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))
 
24-Sep-2022mpbiran2d 440 Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)
(𝜑 → 𝜃)    &   (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   (𝜑 → (𝜓 ↔ 𝜒))
 
24-Sep-2022anim1ci 339 Introduce conjunct to both sides of an implication. (Contributed by Peter Mazsa, 24-Sep-2022.)
(𝜑 → 𝜓)    ⇒   ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓))
 
23-Sep-2022fisumss 11355 Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
 
21-Sep-2022isumss 11354 Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
 
21-Sep-2022pw1dom2 7204 The power set of 1o dominates 2o. Also see pwpw0ss 3791 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
2o ≼ 𝒫 1o
 
18-Sep-2022sumfct 11337 A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 18-Sep-2022.)
(∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵)
 
18-Sep-2022syl21anbrc 1177 Syllogism inference. (Contributed by Peter Mazsa, 18-Sep-2022.)
(𝜑 → 𝜓)    &   (𝜑 → 𝜒)    &   (𝜑 → 𝜃)    &   (𝜏 ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))    ⇒   (𝜑 → 𝜏)
 
18-Sep-2022an21 468 Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.)
(((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
 
16-Sep-2022fser0const 10472 Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.)
𝑍 = (ℤ≥‘𝑀)    ⇒   (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑍 × {0}))
 
8-Sep-2022zfz1isolemiso 10774 Lemma for zfz1iso 10776. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
(𝜑 → 𝑋 ∈ Fin)    &   (𝜑 → 𝑋 ⊆ ℤ)    &   (𝜑 → 𝑀 ∈ 𝑋)    &   (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    &   (𝜑 → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))    &   (𝜑 → 𝐴 ∈ (1...(♯‘𝑋)))    &   (𝜑 → 𝐵 ∈ (1...(♯‘𝑋)))    ⇒   (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
 
8-Sep-2022zfz1isolemsplit 10773 Lemma for zfz1iso 10776. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)
(𝜑 → 𝑋 ∈ Fin)    &   (𝜑 → 𝑀 ∈ 𝑋)    ⇒   (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
 
7-Sep-2022zfz1isolem1 10775 Lemma for zfz1iso 10776. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
(𝜑 → 𝐾 ∈ ω)    &   (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))    &   (𝜑 → 𝑋 ⊆ ℤ)    &   (𝜑 → 𝑋 ∈ Fin)    &   (𝜑 → 𝑋 ≈ suc 𝐾)    &   (𝜑 → 𝑀 ∈ 𝑋)    &   (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    ⇒   (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
 
6-Sep-2022fimaxq 10762 A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)
((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
5-Sep-2022fimax2gtri 6879 A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)
 
5-Sep-2022fimax2gtrilemstep 6878 Lemma for fimax2gtri 6879. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐴 ≠ ∅)    &   (𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝑈 ⊆ 𝐴)    &   (𝜑 → 𝑍 ∈ 𝐴)    &   (𝜑 → 𝑉 ∈ 𝐴)    &   (𝜑 → ¬ 𝑉 ∈ 𝑈)    &   (𝜑 → ∀𝑦 ∈ 𝑈 ¬ 𝑍𝑅𝑦)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
 
5-Sep-2022tridc 6877 A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → 𝐶 ∈ 𝐴)    ⇒   (𝜑 → DECID 𝐵𝑅𝐶)
 
3-Sep-2022zfz1iso 10776 A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
 
2-Sep-2022rspceeqv 2852 Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
(𝑥 = 𝐴 → 𝐶 = 𝐷)    ⇒   ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶)
 
1-Sep-2022ssidd 3168 Weakening of ssid 3167. (Contributed by BJ, 1-Sep-2022.)
(𝜑 → 𝐴 ⊆ 𝐴)
 
31-Aug-2022fveqeq2 5505 Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
(𝐴 = 𝐵 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
 
30-Aug-2022iseqf1olemfvp 10453 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 30-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝑇:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → (⦋𝑇 / 𝑓⦌𝑃‘𝐴) = (𝐺‘(𝑇‘𝐴)))
 
30-Aug-2022fveqeq2d 5504 Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
 
29-Aug-2022seq3f1olemqsumkj 10454 Lemma for seq3f1o 10460. 𝑄 gives the same sum as 𝐽 in the range (𝐾...(◡𝐽‘𝐾)). (Contributed by Jim Kingdon, 29-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    &   (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘(◡𝐽‘𝐾)) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(◡𝐽‘𝐾)))
 
29-Aug-2022iseqf1olemqpcl 10452 Lemma for seq3f1o 10460. A closure lemma involving 𝑄 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
 
29-Aug-2022iseqf1olemjpcl 10451 Lemma for seq3f1o 10460. A closure lemma involving 𝐽 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆)
 
28-Aug-2022iseqf1olemqval 10443 Lemma for seq3f1o 10460. Value of the function 𝑄. (Contributed by Jim Kingdon, 28-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    ⇒   (𝜑 → (𝑄‘𝐴) = if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)))
 
27-Aug-2022iseqf1olemmo 10448 Lemma for seq3f1o 10460. Showing that 𝑄 is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Aug-2022iseqf1olemnanb 10446 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   (𝜑 → ¬ 𝐴 ∈ (𝐾...(◡𝐽‘𝐾)))    &   (𝜑 → ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾)))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Aug-2022iseqf1olemab 10445 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   (𝜑 → 𝐴 ∈ (𝐾...(◡𝐽‘𝐾)))    &   (𝜑 → 𝐵 ∈ (𝐾...(◡𝐽‘𝐾)))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Aug-2022iseqf1olemnab 10444 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    ⇒   (𝜑 → ¬ (𝐴 ∈ (𝐾...(◡𝐽‘𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾))))
 
27-Aug-2022iseqf1olemqcl 10442 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → 𝐴 ∈ (𝑀...𝑁))    ⇒   (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) ∈ (𝑀...𝑁))
 
26-Aug-2022iseqf1olemqf 10447 Lemma for seq3f1o 10460. Domain and codomain of 𝑄. (Contributed by Jim Kingdon, 26-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    ⇒   (𝜑 → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
 
25-Aug-2022fzodcel 10108 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀..^𝑁))
 
24-Aug-2022rspceaimv 2842 Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))
 
22-Aug-2022seq3f1olemqsumk 10455 Lemma for seq3f1o 10460. 𝑄 gives the same sum as 𝐽 in the range (𝐾...𝑁). (Contributed by Jim Kingdon, 22-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    &   (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
 
21-Aug-2022seq3f1olemqsum 10456 Lemma for seq3f1o 10460. 𝑄 gives the same sum as 𝐽. (Contributed by Jim Kingdon, 21-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    &   (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁))
 
21-Aug-2022iseqf1olemqk 10450 Lemma for seq3f1o 10460. 𝑄 is constant for one more position than 𝐽 is. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    ⇒   (𝜑 → ∀𝑥 ∈ (𝑀...𝐾)(𝑄‘𝑥) = 𝑥)
 
21-Aug-2022iseqf1olemqf1o 10449 Lemma for seq3f1o 10460. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))    ⇒   (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
 
21-Aug-2022iseqf1olemklt 10441 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    &   (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾))    ⇒   (𝜑 → 𝐾 < (◡𝐽‘𝐾))
 
21-Aug-2022iseqf1olemkle 10440 Lemma for seq3f1o 10460. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    ⇒   (𝜑 → 𝐾 ≤ (◡𝐽‘𝐾))
 
21-Aug-2022fssdm 5362 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
𝐷 ⊆ dom 𝐹    &   (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → 𝐷 ⊆ 𝐴)
 
21-Aug-2022fssdmd 5361 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐷 ⊆ dom 𝐹)    ⇒   (𝜑 → 𝐷 ⊆ 𝐴)
 
21-Aug-2022eqelssd 3166 Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
21-Aug-2022reximssdv 2574 Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴 ⊆ 𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑 → ∃𝑥 ∈ 𝐵 𝜓)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝑥 ∈ 𝐴)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)
 
21-Aug-2022animpimp2impd 554 Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂)))    &   ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏))    ⇒   (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))
 
20-Aug-2022brimralrspcev 4048 Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓))
 
20-Aug-2022brralrspcev 4047 Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 𝐴𝑅𝐵) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴𝑅𝑥)
 
19-Aug-2022seq3f1olemstep 10457 Lemma for seq3f1o 10460. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)    &   (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
 
18-Aug-2022seq3f1olemp 10458 Lemma for seq3f1o 10460. Existence of a constant permutation of (𝑀...𝑁) which leads to the same sum as the permutation 𝐹 itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))    &   𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
 
17-Aug-2022seq3f1oleml 10459 Lemma for seq3f1o 10460. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))    ⇒   (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
17-Aug-2022imbrov2fvoveq 5878 Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
(𝑋 = 𝑌 → (𝜑 ↔ 𝜓))    ⇒   (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))
 
16-Aug-2022fmpttd 5651 Version of fmptd 5650 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
 
15-Aug-2022nnf1o 11339 Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)
(𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴)    ⇒   (𝜑 → 𝑁 = 𝑀)
 
14-Aug-20222fveq3 5501 Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵)))
 
13-Aug-2022exmidsbth 14056 The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6944) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6944.

The reverse direction (exmidsbthr 14055) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)

(EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦))
 
13-Aug-2022fv0p1e1 8993 Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
(𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))
 
13-Aug-2022ovanraleqv 5877 Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
(𝐵 = 𝑋 → (𝜑 ↔ 𝜓))    ⇒   (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
 
12-Aug-2022stbid 827 The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (STAB 𝜓 ↔ STAB 𝜒))
 
11-Aug-2022exmidsbthr 14055 The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
(∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)
 
11-Aug-2022exmidsbthrlem 14054 Lemma for exmidsbthr 14055. (Contributed by Jim Kingdon, 11-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))    ⇒   (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)
 
10-Aug-2022nninfomni 14052 ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ∞ ∈ Omni
 
10-Aug-2022nninfomnilem 14051 Lemma for nninfomni 14052. (Contributed by Jim Kingdon, 10-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    ⇒   ℕ∞ ∈ Omni
 
10-Aug-2022nninfex 7098 ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ∞ ∈ V
 
10-Aug-2022vpwex 4165 Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4166 from vpwex 4165. (Revised by BJ, 10-Aug-2022.)
𝒫 𝑥 ∈ V
 
9-Aug-2022nninfsel 14050 𝐸 is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)    ⇒   (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o)
 
9-Aug-2022nninfsellemeqinf 14049 Lemma for nninfsel 14050. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)    ⇒   (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o))
 
9-Aug-2022nninfsellemqall 14048 Lemma for nninfsel 14050. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)    &   (𝜑 → 𝑁 ∈ ω)    ⇒   (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = 1o)
 
9-Aug-2022nninfsellemeq 14047 Lemma for nninfsel 14050. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)    &   (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅)    ⇒   (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
 
8-Aug-2022nninfsellemcl 14044 Lemma for nninfself 14046. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → if(∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
 
8-Aug-2022nninfsellemdc 14043 Lemma for nninfself 14046. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
 
8-Aug-2022ss1oel2o 14026 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4184 which more directly illustrates the contrast with el2oss1o 6422. (Contributed by Jim Kingdon, 8-Aug-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o))
 
8-Aug-2022el2oss1o 6422 Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 14026. (Contributed by Jim Kingdon, 8-Aug-2022.)
(𝐴 ∈ 2o → 𝐴 ⊆ 1o)
 
7-Aug-2022nnti 14027 Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
(𝜑 → 𝐴 ∈ ω)    ⇒   ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢)))
 
6-Aug-2022nninfself 14046 Domain and range of the selection function for ℕ∞. (Contributed by Jim Kingdon, 6-Aug-2022.)
𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))    ⇒   𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞
 
6-Aug-2022nninfsellemsuc 14045 Lemma for nninfself 14046. (Contributed by Jim Kingdon, 6-Aug-2022.)
((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝐽 ∈ ω) → if(∀𝑘 ∈ suc suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ⊆ if(∀𝑘 ∈ suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))
 
4-Aug-2022nnnninfeq2 7105 Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7104 but if we have information about a single 1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝜑 → 𝑃 ∈ ℕ∞)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → (𝑃‘∪ 𝑁) = 1o)    &   (𝜑 → (𝑃‘𝑁) = ∅)    ⇒   (𝜑 → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
 
4-Aug-2022nnnninfeq 7104 Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝜑 → 𝑃 ∈ ℕ∞)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → ∀𝑥 ∈ 𝑁 (𝑃‘𝑥) = 1o)    &   (𝜑 → (𝑃‘𝑁) = ∅)    ⇒   (𝜑 → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
 
4-Aug-2022nninff 7099 An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o)
 
1-Aug-2022nninfall 14042 Given a decidable predicate on ℕ∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which 𝑄 is a decidable predicate is that it assigns a value of either ∅ or 1o (which can be thought of as false and true) to every element of ℕ∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)    ⇒   (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o)
 
1-Aug-2022nninfalllem1 14041 Lemma for nninfall 14042. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)    &   (𝜑 → 𝑃 ∈ ℕ∞)    &   (𝜑 → (𝑄‘𝑃) = ∅)    ⇒   (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o)
 
1-Aug-2022peano3nninf 14040 The successor function on ℕ∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))    ⇒   (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
 
31-Jul-2022peano4nninf 14039 The successor function on ℕ∞ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))    ⇒   𝑆:ℕ∞–1-1→ℕ∞
 
31-Jul-20221lt2o 6421 Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
1o ∈ 2o
 
31-Jul-20220lt2o 6420 Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
∅ ∈ 2o
 
31-Jul-2022nnpredcl 4607 The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4548) but also holds when it is ∅ by uni0 3823. (Contributed by Jim Kingdon, 31-Jul-2022.)
(𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
 
31-Jul-2022nnsucpred 4601 The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = 𝐴)
 
30-Jul-2022nnsf 14038 Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))    ⇒   𝑆:ℕ∞⟶ℕ∞
 
29-Jul-2022fodjuomnilemres 7124 Lemma for fodjuomni 7125. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
(𝜑 → 𝑂 ∈ Omni)    &   (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    ⇒   (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))
 
28-Jul-2022eqifdc 3560 Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
(DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))
 
27-Jul-2022fodjuomni 7125 A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
(𝜑 → 𝑂 ∈ Omni)    &   (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    ⇒   (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))
 
27-Jul-2022fodjuomnilemdc 7120 Lemma for fodjuomni 7125. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
(𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))
 
25-Jul-2022djudom 7070 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐷))
 
23-Jul-2022fvoveq1 5876 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5875. (Contributed by AV, 23-Jul-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
23-Jul-2022fvoveq1d 5875 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
17-Jul-2022inftonninf 10397 The mapping of +∞ into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
 
17-Jul-20221tonninf 10396 The mapping of one into ℕ∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
 
17-Jul-20220tonninf 10395 The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   (𝐼‘0) = (𝑥 ∈ ω ↦ ∅)
 
16-Jul-2022fxnn0nninf 10394 A function from ℕ0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7100 instead of infnninfOLD 7101. More generally, this theorem and most theorems in this section could use an extended 𝐺 defined by 𝐺 = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ ⟨ω, +∞⟩) and 𝐹 = (𝑛 ∈ suc ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) as in nnnninf2 7103.
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   𝐼:ℕ0*⟶ℕ∞
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