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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 13-Oct-2025 at 6:51 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
6-Oct-2025dvconstss 14877 Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
(𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   𝐽 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑 → 𝑋 ∈ 𝐽)    &   (𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))
 
3-Oct-2025dvidre 14876 Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(ℝ D ( I ↾ ℝ)) = (ℝ × {1})
 
3-Oct-2025dvconstre 14875 Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
 
3-Oct-2025dvidsslem 14872 Lemma for dvconstss 14877. Analogue of dvidlemap 14870 where 𝐹 is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   𝐽 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ∈ 𝐽)    &   ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   𝐵 ∈ ℂ    ⇒   (𝜑 → (𝑆 D 𝐹) = (𝑋 × {𝐵}))
 
3-Oct-2025dvidrelem 14871 Lemma for dvidre 14876 and dvconstre 14875. Analogue of dvidlemap 14870 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝜑 → 𝐹:ℝ⟶ℂ)    &   ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   𝐵 ∈ ℂ    ⇒   (𝜑 → (ℝ D 𝐹) = (ℝ × {𝐵}))
 
28-Sep-2025metuex 14054 Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
 
28-Sep-2025cndsex 14052 The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(abs ∘ − ) ∈ V
 
25-Sep-2025cntopex 14053 The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
(MetOpen‘(abs ∘ − )) ∈ V
 
24-Sep-2025mopnset 14051 Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
(𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
 
24-Sep-2025blfn 14050 The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
ball Fn V
 
22-Sep-2025plycjlemc 14938 Lemma for plycj 14939. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   (𝜑 → 𝐹 ∈ (Poly‘𝑆))    ⇒   (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘))))
 
20-Sep-2025plycolemc 14936 Lemma for plyco 14937. The result expressed as a sum, with a degree and coefficients for 𝐹 specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)
(𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘))))    ⇒   (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
 
16-Sep-2025lgsquadlemofi 15233 Lemma for lgsquad 15237. There are finitely many members of 𝑆 with odd first part. (Contributed by Jim Kingdon, 16-Sep-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑃 ≠ 𝑄)    &   𝑀 = ((𝑃 − 1) / 2)    &   𝑁 = ((𝑄 − 1) / 2)    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1st ‘𝑧)} ∈ Fin)
 
16-Sep-2025lgsquadlemsfi 15232 Lemma for lgsquad 15237. 𝑆 is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑃 ≠ 𝑄)    &   𝑀 = ((𝑃 − 1) / 2)    &   𝑁 = ((𝑄 − 1) / 2)    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
16-Sep-2025opabfi 6994 Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
13-Sep-2025uchoice 6192 Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7269 (with the key difference being the change of ∃ to ∃!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
 
11-Sep-2025expghmap 14106 Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
𝑀 = (mulGrp‘ℂfld)    &   𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})    ⇒   ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))
 
11-Sep-2025cnfldui 14088 The invertible complex numbers are exactly those apart from zero. This is recapb 8692 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
{𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
 
9-Sep-2025gsumfzfsumlemm 14086 Lemma for gsumfzfsum 14087. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
 
9-Sep-2025gsumfzfsumlem0 14085 Lemma for gsumfzfsum 14087. The case where the sum is empty. (Contributed by Jim Kingdon, 9-Sep-2025.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑁 < 𝑀)    ⇒   (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
 
9-Sep-2025gsumfzmhm2 13417 Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐻 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑋 ∈ 𝐵)    &   (𝑥 = 𝑋 → 𝐶 = 𝐷)    &   (𝑥 = (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   (𝜑 → (𝐻 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)
 
8-Sep-2025gsumfzmhm 13416 Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐻 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
 
6-Sep-2025gsumfzconst 13414 Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
 
31-Aug-2025gsumfzmptfidmadd 13412 The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)    &   𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)    &   𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)    ⇒   (𝜑 → (𝐺 Σg (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
30-Aug-2025gsumfzsubmcl 13411 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
(𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝑆)    ⇒   (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
 
30-Aug-2025seqm1g 10548 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 30-Aug-2025.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   (𝜑 → + ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑊)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)))
 
29-Aug-2025seqf1og 10595 Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 29-Aug-2025.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐶 ⊆ 𝑆)    &   (𝜑 → + ∈ 𝑉)    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))    &   (𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝐻 ∈ 𝑋)    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
25-Aug-2025irrmulap 9716 The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 9715. (Contributed by Jim Kingdon, 25-Aug-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ∀𝑞 ∈ ℚ 𝐴 # 𝑞)    &   (𝜑 → 𝐵 ∈ ℚ)    &   (𝜑 → 𝐵 ≠ 0)    &   (𝜑 → 𝑄 ∈ ℚ)    ⇒   (𝜑 → (𝐴 · 𝐵) # 𝑄)
 
19-Aug-2025seqp1g 10540 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐹 ∈ 𝑉 ∧ + ∈ 𝑊) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
19-Aug-2025seq1g 10537 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉 ∧ + ∈ 𝑊) → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
 
18-Aug-2025iswrdiz 10924 A zero-based sequence is a word. In iswrdinn0 10922 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)
((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℤ) → 𝑊 ∈ Word 𝑆)
 
16-Aug-2025gsumfzcl 13074 Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
16-Aug-2025iswrdinn0 10922 A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)
((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℕ0) → 𝑊 ∈ Word 𝑆)
 
15-Aug-2025gsumfzz 13070 Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
 
14-Aug-2025gsumfzval 12977 An expression for Σg when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
 
13-Aug-2025znidom 14156 The ℤ/nℤ structure is an integral domain when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.)
𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℙ → 𝑌 ∈ IDomn)
 
12-Aug-2025rrgmex 13760 A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
𝐸 = (RLReg‘𝑅)    ⇒   (𝐴 ∈ 𝐸 → 𝑅 ∈ V)
 
10-Aug-2025gausslemma2dlem1cl 15216 Lemma for gausslemma2dlem1 15218. Closure of the body of the definition of 𝑅. (Contributed by Jim Kingdon, 10-Aug-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   𝐻 = ((𝑃 − 1) / 2)    &   𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   (𝜑 → 𝐴 ∈ (1...𝐻))    ⇒   (𝜑 → if((𝐴 · 2) < (𝑃 / 2), (𝐴 · 2), (𝑃 − (𝐴 · 2))) ∈ ℤ)
 
9-Aug-2025gausslemma2dlem1f1o 15217 Lemma for gausslemma2dlem1 15218. (Contributed by Jim Kingdon, 9-Aug-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   𝐻 = ((𝑃 − 1) / 2)    &   𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    ⇒   (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
 
7-Aug-2025qdclt 10318 Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
 
22-Jul-2025ivthdich 14832 The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 14822 for strictly monotonic functions, can be proved).

The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number 𝑧. We want to show that 𝑧 ≤ 0 ∨ 0 ≤ 𝑧. Because of hovercncf 14825, hovera 14826, and hoverb 14827, we are able to apply the intermediate value theorem to get a value 𝑐 such that the hover function at 𝑐 equals 𝑧. By axltwlin 8089, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 14828 or 0 ≤ 𝑧 by hovergt0 14829. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)

(∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ∀𝑟 ∈ ℝ ∀𝑠 ∈ ℝ (𝑟 ≤ 𝑠 ∨ 𝑠 ≤ 𝑟))
 
22-Jul-2025dich0 14831 Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
(∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
 
22-Jul-2025ivthdichlem 14830 Lemma for ivthdich 14832. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    &   (𝜑 → 𝑍 ∈ ℝ)    &   (𝜑 → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))    ⇒   (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))
 
22-Jul-2025hovergt0 14829 The hover function evaluated at a point greater than zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → 0 ≤ (𝐹‘𝐶))
 
22-Jul-2025hoverlt1 14828 The hover function evaluated at a point less than one. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   ((𝐶 ∈ ℝ ∧ 𝐶 < 1) → (𝐹‘𝐶) ≤ 0)
 
21-Jul-2025hoverb 14827 A point at which the hover function is greater than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   (𝑍 ∈ ℝ → 𝑍 < (𝐹‘(𝑍 + 2)))
 
21-Jul-2025hovera 14826 A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
 
21-Jul-2025rexeqtrrdv 2701 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrrdv 2700 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025rexeqtrdv 2699 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrdv 2698 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
 
20-Jul-2025hovercncf 14825 The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   𝐹 ∈ (ℝ–cn→ℝ)
 
19-Jul-2025mincncf 14795 The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℝ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℝ))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ inf({𝐴, 𝐵}, ℝ, < )) ∈ (𝑋–cn→ℝ))
 
18-Jul-2025maxcncf 14794 The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℝ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℝ))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ sup({𝐴, 𝐵}, ℝ, < )) ∈ (𝑋–cn→ℝ))
 
14-Jul-2025xnn0nnen 10511 The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
ℕ0* ≈ ℕ
 
12-Jul-2025nninfninc 7184 All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
(𝜑 → 𝐴 ∈ ℕ∞)    &   (𝜑 → 𝑋 ∈ ω)    &   (𝜑 → 𝑌 ∈ ω)    &   (𝜑 → 𝑋 ⊆ 𝑌)    &   (𝜑 → (𝐴‘𝑋) = ∅)    ⇒   (𝜑 → (𝐴‘𝑌) = ∅)
 
10-Jul-2025nninfctlemfo 12180 Lemma for nninfct 12181. (Contributed by Jim Kingdon, 10-Jul-2025.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   (ω ∈ Omni → 𝐼:ℕ0*–onto→ℕ∞)
 
8-Jul-2025nnnninfen 15581 Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
(ω ≈ ℕ∞ ↔ ω ∈ Omni)
 
8-Jul-2025nninfct 12181 The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
(ω ∈ Omni → ∃𝑓 𝑓:ω–onto→(ℕ∞ ⊔ 1o))
 
8-Jul-2025nninfinf 10517 ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
ω ≼ ℕ∞
 
7-Jul-2025ivthreinc 14824 Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 14822). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function 𝐹 is continuous on the entire real line, not just (𝐴[,]𝐵) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (𝜑 → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))    ⇒   (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
28-Jun-2025fngsum 12974 Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
Σg Fn (V × V)
 
28-Jun-2025iotaexel 5879 Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)
 
27-Jun-2025df-igsum 12873 Define a finite group sum (also called "iterated sum") of a structure.

Given 𝐺 Σg 𝐹 where 𝐹:𝐴⟶(Base‘𝐺), the set of indices is 𝐴 and the values are given by 𝐹 at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺.

1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity.

2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ((𝐹‘1) + (𝐹‘2)) + (𝐹‘3), etc.

3. This definition does not handle other cases.

(Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)

Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ (℩𝑥((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))))
 
20-Jun-2025opprnzrbg 13684 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 13685. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
 
16-Jun-2025fnpsr 14164 The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
mPwSer Fn (V × V)
 
14-Jun-2025basm 12682 A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝐵 → ∃𝑗 𝑗 ∈ 𝐺)
 
14-Jun-2025elfvm 5588 If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
(𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)
 
6-Jun-2025pcxqcl 12453 The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → ((𝑃 pCnt 𝑁) ∈ ℤ ∨ (𝑃 pCnt 𝑁) = +∞))
 
5-Jun-2025xqltnle 10339 "Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +∞. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in ℕ0* or ℝ*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
(((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
 
5-Jun-2025ceqsexv2d 2800 Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜓    ⇒   ∃𝑥𝜑
 
30-May-20254sqexercise2 12540 Exercise which may help in understanding the proof of 4sqlemsdc 12541. (Contributed by Jim Kingdon, 30-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑛 = ((𝑥↑2) + (𝑦↑2))}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
27-May-2025iotaexab 5234 Existence of the ℩ class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)
({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)
 
25-May-20254sqlemsdc 12541 Lemma for 4sq 12551. The property of being the sum of four squares is decidable.

The proof involves showing that (for a particular 𝐴) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 12539 and 4sqexercise2 12540 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)

𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
25-May-20254sqexercise1 12539 Exercise which may help in understanding the proof of 4sqlemsdc 12541. (Contributed by Jim Kingdon, 25-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
24-May-20254sqleminfi 12538 Lemma for 4sq 12551. 𝐴 ∩ ran 𝐹 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))    ⇒   (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)
 
24-May-20254sqlemffi 12537 Lemma for 4sq 12551. ran 𝐹 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))    ⇒   (𝜑 → ran 𝐹 ∈ Fin)
 
24-May-20254sqlemafi 12536 Lemma for 4sq 12551. 𝐴 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    ⇒   (𝜑 → 𝐴 ∈ Fin)
 
24-May-2025infidc 6995 The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) ∈ Fin)
 
19-May-2025zrhex 14120 Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
𝐿 = (ℤRHom‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐿 ∈ V)
 
16-May-2025rhmex 13656 Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RingHom 𝑆) ∈ V)
 
15-May-2025ghmex 13328 The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) ∈ V)
 
15-May-2025mhmex 13037 The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)
 
14-May-2025idomcringd 13777 An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
6-May-2025rrgnz 13767 In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
𝐸 = (RLReg‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸)
 
5-May-2025rngressid 13453 A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 12692. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)
 
5-May-2025ablressid 13408 A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12692. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
 
29-Apr-2025rlmscabas 13959 Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
(𝑅 ∈ 𝑋 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅))))
 
29-Apr-2025ressbasid 12691 The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
𝐵 = (Base‘𝑊)    ⇒   (𝑊 ∈ 𝑉 → (Base‘(𝑊 ↾s 𝐵)) = 𝐵)
 
28-Apr-2025lssmex 13854 If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
𝑆 = (LSubSp‘𝑊)    ⇒   (𝑈 ∈ 𝑆 → 𝑊 ∈ V)
 
27-Apr-2025cnfldmul 14063 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
· = (.r‘ℂfld)
 
27-Apr-2025cnfldadd 14061 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
+ = (+g‘ℂfld)
 
27-Apr-2025lidlex 13972 Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
 
27-Apr-2025lssex 13853 Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)
 
25-Apr-2025rspex 13973 Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
 
25-Apr-2025lspex 13894 Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑋 → (LSpan‘𝑊) ∈ V)
 
25-Apr-2025eqgex 13294 The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)
 
25-Apr-2025qusex 12911 Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)
 
23-Apr-20251dom1el 15553 If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 = 𝐶)
 
22-Apr-2025mulgex 13196 Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
(𝐺 ∈ 𝑉 → (.g‘𝐺) ∈ V)
 
20-Apr-2025elovmpod 6118 Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6119 in deduction form. (Revised by AV, 20-Apr-2025.)
𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))
 
18-Apr-2025df2idl2 14008 Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
 
18-Apr-20252idlmex 14000 Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
𝑇 = (2Ideal‘𝑊)    ⇒   (𝑈 ∈ 𝑇 → 𝑊 ∈ V)
 
18-Apr-2025dflidl2 13987 Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
 
18-Apr-2025lidlmex 13974 Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
𝐼 = (LIdeal‘𝑊)    ⇒   (𝑈 ∈ 𝐼 → 𝑊 ∈ V)
 
18-Apr-2025lsslsp 13928 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
𝑋 = (𝑊 ↾s 𝑈)    &   𝑀 = (LSpan‘𝑊)    &   𝑁 = (LSpan‘𝑋)    &   𝐿 = (LSubSp‘𝑊)    ⇒   ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) = (𝑀‘𝐺))
 
16-Apr-2025sraex 13945 Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → 𝐴 ∈ V)
 
14-Apr-2025grpmgmd 13101 A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mgm)
 
12-Apr-2025psraddcl 14175 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g‘𝑆)    &   (𝜑 → 𝑅 ∈ Mgm)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
10-Apr-2025cndcap 15619 Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
 
4-Apr-2025ghmf1 13346 Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g‘𝑅)    &    0 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))
 
3-Apr-2025quscrng 14032 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)    ⇒   ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
 
31-Mar-2025cnfldds 14067 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14056. (Revised by GG, 31-Mar-2025.)
(abs ∘ − ) = (dist‘ℂfld)
 
31-Mar-2025cnfldle 14066 The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14056. (Revised by GG, 31-Mar-2025.)
≤ = (le‘ℂfld)
 
31-Mar-2025cnfldtset 14065 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
(MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
 
31-Mar-2025mpocnfldmul 14062 The multiplication operation of the field of complex numbers. Version of cnfldmul 14063 using maps-to notation, which does not require ax-mulf 7997. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
 
31-Mar-2025mpocnfldadd 14060 The addition operation of the field of complex numbers. Version of cnfldadd 14061 using maps-to notation, which does not require ax-addf 7996. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
 
31-Mar-2025df-cnfld 14056 The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator.

The contract of this set is defined entirely by cnfldex 14058, cnfldadd 14061, cnfldmul 14063, cnfldcj 14064, cnfldtset 14065, cnfldle 14066, cnfldds 14067, and cnfldbas 14059. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)

ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
 
31-Mar-20252idlcpbl 14023 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
22-Mar-2025idomringd 13778 An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
22-Mar-2025idomdomd 13776 An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Domn)
 
21-Mar-2025df2idl2rng 14007 Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
 
21-Mar-2025isridlrng 13981 A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
 
21-Mar-2025dflidl2rng 13980 Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼))
 
20-Mar-2025ccoslid 12852 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
(comp = Slot (comp‘ndx) ∧ (comp‘ndx) ∈ ℕ)
 
20-Mar-2025homslid 12850 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
(Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
 
19-Mar-2025ptex 12878 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
(𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)
 
18-Mar-2025prdsex 12883 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
 
16-Mar-2025plycn 14940 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 7997. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
16-Mar-2025expcn 14748 The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 7997. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025mpomulcn 14745 Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
16-Mar-2025mpomulf 8011 Multiplication is an operation on complex numbers. Version of ax-mulf 7997 using maps-to notation, proved from the axioms of set theory and ax-mulcl 7972. (Contributed by GG, 16-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
 
13-Mar-20252idlss 14013 A two-sided ideal is a subset of the base set. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
𝐵 = (Base‘𝑊)    &   𝐼 = (2Ideal‘𝑊)    ⇒   (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
 
13-Mar-2025imasex 12891 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)
 
11-Mar-2025rng2idlsubgsubrng 14019 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rng2idlsubrng 14016 A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rnglidlrng 13997 A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
 
11-Mar-2025rnglidlmsgrp 13996 The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
 
11-Mar-2025rnglidlmmgm 13995 The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
 
11-Mar-2025imasival 12892 Value of an image structure. The is a lemma for the theorems imasbas 12893, imasplusg 12894, and imasmulr 12895 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    × = (.r‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})    &   (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   (𝜑 → 𝑅 ∈ 𝑍)    ⇒   (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
 
9-Mar-20252idlridld 14006 A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝑂 = (oppr‘𝑅)    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
 
9-Mar-20252idllidld 14005 A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
 
9-Mar-2025quseccl 13306 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)
 
9-Mar-2025fovcl 6025 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
8-Mar-2025subgex 13249 The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
(𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)
 
7-Mar-2025ringrzd 13545 The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 0 ) = 0 )
 
7-Mar-2025ringlzd 13544 The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 · 𝑋) = 0 )
 
7-Mar-2025qusecsub 13404 Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
𝐵 = (Base‘𝐺)    &    − = (-g‘𝐺)    &    ∼ = (𝐺 ~QG 𝑆)    ⇒   (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))
 
1-Mar-2025quselbasg 13303 Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝑈 = (𝐺 /s ∼ )    &   𝐵 = (Base‘𝐺)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
 
28-Feb-2025qusmulrng 14031 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14032. Similar to qusmul2 14028. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
∼ = (𝑅 ~QG 𝑆)    &   𝐻 = (𝑅 /s ∼ )    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    ∙ = (.r‘𝐻)    ⇒   (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
 
28-Feb-2025ringressid 13562 A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12692. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)
 
28-Feb-2025grpressid 13136 A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12692. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
 
27-Feb-2025imasringf1 13564 The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring)
 
26-Feb-2025strext 12726 Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐶 we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
(𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   (𝜑 → 𝐶 ∈ (ℤ≥‘𝐵))    ⇒   (𝜑 → 𝐹 Struct ⟨𝐴, 𝐶⟩)
 
25-Feb-2025subrngringnsg 13704 A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
 
25-Feb-2025rngansg 13449 Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
(𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
 
25-Feb-2025ecqusaddd 13311 Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ))
 
24-Feb-2025ecqusaddcl 13312 Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
 
24-Feb-2025quseccl0g 13304 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13306 for arbitrary sets 𝐺. (Revised by AV, 24-Feb-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝐻 = (𝐺 /s ∼ )    &   𝐶 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶 ∧ 𝑆 ∈ 𝑍) → [𝑋] ∼ ∈ 𝐵)
 
23-Feb-2025ltlenmkv 15630 If < can be expressed as holding exactly when ≤ holds and the values are not equal, then the analytic Markov's Principle applies. (To get the regular Markov's Principle, combine with neapmkv 15628). (Contributed by Jim Kingdon, 23-Feb-2025.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦))
 
23-Feb-2025neap0mkv 15629 The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) ↔ ∀𝑥 ∈ ℝ (𝑥 ≠ 0 → 𝑥 # 0))
 
23-Feb-2025qus2idrng 14024 The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14026 analog). (Contributed by AV, 23-Feb-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
 
23-Feb-20252idlcpblrng 14022 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
23-Feb-2025lringuplu 13695 If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → 𝑅 ∈ LRing)    &   (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
 
23-Feb-2025lringnz 13694 A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
1 = (1r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ LRing → 1 ≠ 0 )
 
23-Feb-2025lringring 13693 A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ Ring)
 
23-Feb-2025lringnzr 13692 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ NzRing)
 
23-Feb-2025islring 13691 The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    1 = (1r‘𝑅)    &   𝑈 = (Unit‘𝑅)    ⇒   (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
 
23-Feb-2025df-lring 13690 A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
 
23-Feb-202501eq0ring 13688 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
 
23-Feb-2025nzrring 13682 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 
23-Feb-2025qusrng 13457 The quotient structure of a non-unital ring is a non-unital ring (qusring2 13565 analog). (Contributed by AV, 23-Feb-2025.)
(𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → ∼ Er 𝑉)    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
23-Feb-2025rngsubdir 13451 Ring multiplication distributes over subtraction. (subdir 8407 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 13556. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍)))
 
23-Feb-2025rngsubdi 13450 Ring multiplication distributes over subtraction. (subdi 8406 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 13555. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍)))
 
22-Feb-2025imasrngf1 13456 The image of a non-unital ring under an injection is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Rng) → 𝑈 ∈ Rng)
 
22-Feb-2025imasrng 13455 The image structure of a non-unital ring is a non-unital ring (imasring 13563 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
22-Feb-2025rngmgpf 13436 Restricted functionality of the multiplicative group on non-unital rings (mgpf 13510 analog). (Contributed by AV, 22-Feb-2025.)
(mulGrp ↾ Rng):Rng⟶Smgrp
 
22-Feb-2025imasabl 13409 The image structure of an abelian group is an abelian group (imasgrp 13184 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑅 ∈ Abel)    &    0 = (0g‘𝑅)    ⇒   (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
 
21-Feb-2025dftap2 7313 Tight apartness with the apartness properties from df-pap 7310 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
(𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
 
20-Feb-2025rng2idlsubg0 14021 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlsubgnsg 14020 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-2025rng2idl0 14018 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlnsg 14017 A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-20252idlelbas 14015 The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
 
20-Feb-20252idlbas 14014 The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → 𝐵 = 𝐼)
 
20-Feb-20252idlelb 14004 Membership in a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)    ⇒   (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
 
20-Feb-2025aprap 13785 The relation given by df-apr 13780 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))
 
20-Feb-2025setscomd 12662 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝐵 ∈ 𝑍)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝑊)    &   (𝜑 → 𝐷 ∈ 𝑋)    ⇒   (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
 
20-Feb-2025ifnebibdc 3601 The converse of ifbi 3578 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))
 
20-Feb-2025ifnefals 3600 Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
 
20-Feb-2025ifnetruedc 3599 Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
 
18-Feb-2025rnglidlmcl 13979 A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
 
17-Feb-2025aprcotr 13784 The apartness relation given by df-apr 13780 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ LRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 → (𝑋 # 𝑍 ∨ 𝑌 # 𝑍)))
 
17-Feb-2025aprsym 13783 The apartness relation given by df-apr 13780 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 → 𝑌 # 𝑋))
 
17-Feb-2025aprval 13781 Expand Definition df-apr 13780. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → − = (-g‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 ↔ (𝑋 − 𝑌) ∈ 𝑈))
 
17-Feb-2025subrngpropd 13715 If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))
 
17-Feb-2025rngm2neg 13448 Double negation of a product in a non-unital ring (mul2neg 8419 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 13554. (Revised by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
 
17-Feb-2025rngmneg2 13447 Negation of a product in a non-unital ring (mulneg2 8417 analog). In contrast to ringmneg2 13553, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌)))
 
17-Feb-2025rngmneg1 13446 Negation of a product in a non-unital ring (mulneg1 8416 analog). In contrast to ringmneg1 13552, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
 
16-Feb-2025aprirr 13782 The apartness relation given by df-apr 13780 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))    ⇒   (𝜑 → ¬ 𝑋 # 𝑋)
 
16-Feb-2025rngrz 13445 The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 13543. (Revised by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
16-Feb-2025rng0cl 13442 The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Rng → 0 ∈ 𝐵)
 
16-Feb-2025rngacl 13441 Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
16-Feb-2025rnggrp 13437 A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
(𝑅 ∈ Rng → 𝑅 ∈ Grp)
 
16-Feb-2025aptap 8671 Complex apartness (as defined at df-ap 8603) is a tight apartness (as defined at df-tap 7312). (Contributed by Jim Kingdon, 16-Feb-2025.)
# TAp ℂ
 
15-Feb-2025subsubrng2 13714 The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))
 
15-Feb-2025subsubrng 13713 A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
 
15-Feb-2025subrngin 13712 The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRng‘𝑅))
 
15-Feb-2025subrngintm 13711 The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))
 
15-Feb-2025opprsubrngg 13710 Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))
 
15-Feb-2025issubrng2 13709 Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
 
15-Feb-2025opprrngbg 13577 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13576. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
 
15-Feb-2025opprrng 13576 An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝑂 ∈ Rng)
 
15-Feb-2025rngpropd 13454 If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
 
15-Feb-2025sgrppropd 12999 If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐿 ∈ 𝑊)    &   (𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
 
15-Feb-2025sgrpcl 12995 Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝐺)    &    ⚬ = (+g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
 
15-Feb-2025tapeq2 7315 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
(𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
 
14-Feb-2025subrngmcl 13708 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 13732. (Revised by AV, 14-Feb-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
14-Feb-2025subrngacl 13707 A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
+ = (+g‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
 
14-Feb-2025subrng0 13706 A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    &    0 = (0g‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 0 = (0g‘𝑆))
 
14-Feb-2025subrngbas 13705 Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
 
14-Feb-2025subrngsubg 13703 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
14-Feb-2025subrngrcl 13702 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
 
14-Feb-2025subrngrng 13701 A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
 
14-Feb-2025subrngid 13700 Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))
 
14-Feb-2025subrngss 13699 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
 
14-Feb-2025issubrng 13698 The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
 
14-Feb-2025df-subrng 13697 Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
 
14-Feb-2025isrngd 13452 Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → · = (.r‘𝑅))    &   (𝜑 → 𝑅 ∈ Abel)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    ⇒   (𝜑 → 𝑅 ∈ Rng)
 
14-Feb-2025rngdi 13439 Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
14-Feb-2025exmidmotap 7323 The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
 
14-Feb-2025exmidapne 7322 Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))
 
14-Feb-2025df-pap 7310 Apartness predicate. A relation 𝑅 is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
(𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
 
13-Feb-20252idl1 14012 Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
 
13-Feb-20252idl0 14011 Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
 
13-Feb-2025ridl1 14010 Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
 
13-Feb-2025ridl0 14009 Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
 
13-Feb-2025isridl 14003 A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
 
13-Feb-2025df-apr 13780 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 13785. (Contributed by Jim Kingdon, 13-Feb-2025.)
#r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ (𝑥(-g‘𝑤)𝑦) ∈ (Unit‘𝑤))})
 
13-Feb-2025rngass 13438 Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
13-Feb-2025issgrpd 12998 Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐺))    &   (𝜑 → + = (+g‘𝐺))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → 𝐺 ∈ Smgrp)
 
8-Feb-20252oneel 7318 ∅ and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
 
8-Feb-2025tapeq1 7314 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
(𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
 
7-Feb-2025resrhm2b 13748 Restriction of the codomain of a (ring) homomorphism. resghm2b 13335 analog. (Contributed by SN, 7-Feb-2025.)
𝑈 = (𝑇 ↾s 𝑋)    ⇒   ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))
 
6-Feb-2025zzlesq 10782 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
6-Feb-20252omotap 7321 If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
(∃*𝑟 𝑟 TAp 2o → EXMID)
 
6-Feb-20252omotaplemst 7320 Lemma for 2omotap 7321. (Contributed by Jim Kingdon, 6-Feb-2025.)
((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
 
6-Feb-20252omotaplemap 7319 Lemma for 2omotap 7321. (Contributed by Jim Kingdon, 6-Feb-2025.)
(¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
 
6-Feb-20252onetap 7317 Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o
 
5-Feb-2025netap 7316 Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
 
5-Feb-2025df-tap 7312 Tight apartness predicate. A relation 𝑅 is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
(𝑅 TAp 𝐴 ↔ (𝑅 Ap 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))
 
1-Feb-2025mulgnn0cld 13216 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13211. (Contributed by SN, 1-Feb-2025.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
 
31-Jan-20250subg 13272 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
 
29-Jan-2025grprinvd 13131 The right inverse of a group element. Deduction associated with grprinv 13126. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )
 
29-Jan-2025grplinvd 13130 The left inverse of a group element. Deduction associated with grplinv 13125. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )
 
29-Jan-2025grpinvcld 13124 A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁‘𝑋) ∈ 𝐵)
 
29-Jan-2025grpridd 13109 The identity element of a group is a right identity. Deduction associated with grprid 13107. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 0 ) = 𝑋)
 
29-Jan-2025grplidd 13108 The identity element of a group is a left identity. Deduction associated with grplid 13106. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 + 𝑋) = 𝑋)
 
29-Jan-2025grpassd 13087 A group operation is associative. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
28-Jan-2025dvdsrex 13597 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)
 
24-Jan-2025reldvdsrsrg 13591 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
(𝑅 ∈ SRing → Rel (∥r‘𝑅))
 
18-Jan-2025rerecapb 8864 A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ ℝ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
 
18-Jan-2025recapb 8692 A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1))
 
17-Jan-2025ressval3d 12693 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆 ↾s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → 𝐸 ∈ dom 𝑆)    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
17-Jan-2025strressid 12692 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(𝜑 → 𝐵 = (Base‘𝑊))    &   (𝜑 → 𝑊 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑊)    ⇒   (𝜑 → (𝑊 ↾s 𝐵) = 𝑊)
 
16-Jan-2025ressex 12686 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 12685 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
 
12-Jan-2025isrim 13668 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
 
10-Jan-2025rimrhm 13670 A ring isomorphism is a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
 
10-Jan-2025isrim0 13660 A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
 
10-Jan-2025opprex 13572 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 13578. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
 
10-Jan-2025mgpex 13424 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 13467. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
 
5-Jan-2025imbibi 252 The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
(((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))
 
1-Jan-2025snss 3754 The singleton of an element of a class is a subset of the class (inference form of snssg 3753). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
𝐴 ∈ V    ⇒   (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
 
1-Jan-2025snssg 3753 The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
 
1-Jan-2025snssb 3752 Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
 
9-Dec-2024nninfwlpoim 7239 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 7238 Lemma for nninfwlpoim 7239. (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 7237 Lemma for nninfwlpoim 7239. (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 7236 Lemma for nninfwlpoim 7239. (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 7235 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 7234 Lemma for nninfwlpor 7235. 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 7233 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 7240 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 7232 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 12681 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 5244 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5243 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 12680 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝐵 → 𝐺 ∈ V)
 
14-Nov-2024dcand 934 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)
(𝜑 → DECID 𝜓)    &   (𝜑 → DECID 𝜒)    ⇒   (𝜑 → DECID (𝜓 ∧ 𝜒))
 
12-Nov-2024sravscag 13942 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024srascag 13941 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
 
12-Nov-2024slotsdifipndx 12795 The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
(( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
 
11-Nov-2024bj-con1st 15313 Contraposition when the antecedent is a negated stable proposition. See con1dc 857. (Contributed by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
 
11-Nov-2024slotsdifdsndx 12841 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
 
11-Nov-2024slotsdifplendx 12830 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
 
11-Nov-2024tsetndxnstarvndx 12814 The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(TopSet‘ndx) ≠ (*𝑟‘ndx)
 
11-Nov-2024ofeqd 6134 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
11-Nov-2024const 853 Contraposition when the antecedent is a negated stable proposition. See comment of condc 854. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
10-Nov-2024slotsdifunifndx 12848 The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
(((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
 
7-Nov-2024ressbasd 12688 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(𝜑 → 𝑅 = (𝑊 ↾s 𝐴))    &   (𝜑 → 𝐵 = (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
 
6-Nov-2024oppraddg 13575 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &    + = (+g‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → + = (+g‘𝑂))
 
6-Nov-2024opprbasg 13574 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐵 = (Base‘𝑂))
 
6-Nov-2024opprsllem 13573 Lemma for opprbasg 13574 and oppraddg 13575. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (.r‘ndx)    ⇒   (𝑅 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑂))
 
4-Nov-2024lgsfvalg 15162 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))
 
3-Nov-2024znmul 14141 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
 
3-Nov-2024znadd 14140 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
 
3-Nov-2024znbas2 14139 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
 
3-Nov-2024znbaslemnn 14138 Lemma for znbas 14143. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌))
 
3-Nov-2024zlmmulrg 14130 Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → · = (.r‘𝑊))
 
3-Nov-2024zlmplusgg 14129 Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → + = (+g‘𝑊))
 
3-Nov-2024zlmbasg 14128 Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐵 = (Base‘𝑊))
 
3-Nov-2024zlmlemg 14127 Lemma for zlmbasg 14128 and zlmplusgg 14129. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    &   (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)    ⇒   (𝐺 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑊))
 
2-Nov-2024zlmsca 14131 Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊))
 
1-Nov-2024plendxnvscandx 12829 The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ ( ·𝑠 ‘ndx)
 
1-Nov-2024plendxnscandx 12828 The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (Scalar‘ndx)
 
1-Nov-2024plendxnmulrndx 12827 The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (.r‘ndx)
 
1-Nov-2024qsqeqor 10724 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)))
 
31-Oct-2024dsndxnmulrndx 12838 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(dist‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnmulrndx 12813 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnbasendx 12811 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (Base‘ndx)
 
31-Oct-2024basendxlttsetndx 12810 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(Base‘ndx) < (TopSet‘ndx)
 
31-Oct-2024tsetndxnn 12809 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ∈ ℕ
 
30-Oct-2024plendxnbasendx 12825 The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
(le‘ndx) ≠ (Base‘ndx)
 
30-Oct-2024basendxltplendx 12824 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
(Base‘ndx) < (le‘ndx)
 
30-Oct-2024plendxnn 12823 The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
(le‘ndx) ∈ ℕ
 
29-Oct-2024sradsg 13947 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (dist‘𝑊) = (dist‘𝐴))
 
29-Oct-2024sratsetg 13944 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
 
29-Oct-2024sramulrg 13940 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (.r‘𝑊) = (.r‘𝐴))
 
29-Oct-2024sraaddgg 13939 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (+g‘𝑊) = (+g‘𝐴))
 
29-Oct-2024srabaseg 13938 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (Base‘𝑊) = (Base‘𝐴))
 
29-Oct-2024sralemg 13937 Lemma for srabaseg 13938 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
 
29-Oct-2024dsndxntsetndx 12840 The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 12839 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024slotstnscsi 12815 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024ipndxnmulrndx 12794 The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024ipndxnplusgndx 12793 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (+g‘ndx)
 
29-Oct-2024vscandxnmulrndx 12781 The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024scandxnmulrndx 12776 The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024fiubnn 10904 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 10903 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 10902 Lemma for fiubz 10903 and fiubnn 10904. 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)
 
28-Oct-2024unifndxntsetndx 12847 The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ≠ (TopSet‘ndx)
 
28-Oct-2024basendxltunifndx 12845 The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (UnifSet‘ndx)
 
28-Oct-2024unifndxnn 12844 The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ∈ ℕ
 
28-Oct-2024dsndxnbasendx 12836 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
(dist‘ndx) ≠ (Base‘ndx)
 
28-Oct-2024basendxltdsndx 12835 The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 12834 The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
27-Oct-2024bj-nnst 15305 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 15552 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
¬ ¬ STAB 𝜑
 
27-Oct-2024bj-imnimnn 15300 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 15299 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒    ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 12179 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 12176 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 12177 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 15552 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 851, 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)
 
21-Oct-2024unifndxnbasendx 12846 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(UnifSet‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024ipndxnbasendx 12792 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(·𝑖‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024scandxnbasendx 12774 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalar‘ndx) ≠ (Base‘ndx)
 
20-Oct-2024isprm5lem 12282 Lemma for isprm5 12283. 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)
 
19-Oct-2024resseqnbasd 12694 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (Base‘ndx)    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → 𝐶 = (𝐸‘𝑅))
 
18-Oct-2024rmodislmod 13850 The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 13788 of a left module, see also islmod 13790. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    ⨣ = (+g‘𝐹)    &    × = (.r‘𝐹)    &    1 = (1r‘𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   (𝐹 ∈ CRing → 𝐿 ∈ LMod)
 
18-Oct-2024mgpress 13430 Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
𝑆 = (𝑅 ↾s 𝐴)    &   𝑀 = (mulGrp‘𝑅)    ⇒   ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
 
18-Oct-2024dsndxnplusgndx 12837 The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024plendxnplusgndx 12826 The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(le‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024tsetndxnplusgndx 12812 The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(TopSet‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnscandx 12782 The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
 
18-Oct-2024vscandxnplusgndx 12780 The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnbasendx 12779 The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (Base‘ndx)
 
18-Oct-2024scandxnplusgndx 12775 The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(Scalar‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnmulrndx 12764 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (.r‘ndx)
 
18-Oct-2024starvndxnplusgndx 12763 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnbasendx 12762 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (Base‘ndx)
 
17-Oct-2024basendxltplusgndx 12734 The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
(Base‘ndx) < (+g‘ndx)
 
17-Oct-2024plusgndxnn 12732 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+g‘ndx) ∈ ℕ
 
17-Oct-2024elnndc 9680 Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 11389 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 11388 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 12451 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 9872 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
13-Oct-2024xnn0dcle 9871 Decidability of ≤ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → DECID 𝐴 ≤ 𝐵)
 
9-Oct-2024nn0leexp2 10784 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 12429 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 9679 Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 12428 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 12427 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 10783 Special case of ltexp2 15115 which we use here because we haven't yet defined df-rpcxp 15035 which is used in the current proof of ltexp2 15115. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 12095 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7995.) (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 12094 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7995.) (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 12092 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 7095 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 12614 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 12271 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 7042 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 12607 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 12606 Lemma for ssnnct 12607. 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 12093 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 9666 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 12610 Lemma for nninfdc 12613. 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 12175 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 12174. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 12608 Lemma for nninfdc 12613. (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 12611 Lemma for nninfdc 12613. The function from nninfdclemf 12609 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 12613 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 12612 Lemma for nninfdc 12613. The function from nninfdclemf 12609 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 12609 Lemma for nninfdc 12613. 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 12174 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 12605 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 4657 The predecessor (see nnpredcl 4656) of a nonzero natural number is less than (see df-iord 4398) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7193 Lemma for nninfisol 7194. 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 7192 Lemma for nninfisol 7194. 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 7191 Lemma for nninfisol 7194. 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 7194 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)
 
8-Sep-2024relopabv 4787 A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4789. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
7-Sep-2024eulerthlemfi 12369 Lemma for eulerth 12374. 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 10740 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 12372 Lemma for eulerth 12374. 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 12373 Lemma for eulerth 12374. 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 12371 Lemma for eulerth 12374. (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 12370 Lemma for eulerth 12374. 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)
 
1-Sep-2024qusmul2 14028 Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    × = (.r‘𝑄)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
 
30-Aug-2024fprodap0f 11782 A finite product of terms apart from zero is apart from zero. A version of fprodap0 11767 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 11775 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 7305 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 7301 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 4605 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4604. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
 
26-Aug-2024ontriexmidim 4555 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4554. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
 
25-Aug-2024onntri2or 7308 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
25-Aug-2024onntri3or 7307 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 3092 Composition law for chained substitutions into a class. Version of csbco 3091 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-2024.)
⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
 
25-Aug-2024cbvreuvw 2732 Version of cbvreuv 2728 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvrexvw 2731 Version of cbvrexv 2727 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvralvw 2730 Version of cbvralv 2726 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvabw 2316 Version of cbvab 2317 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
 
25-Aug-2024nfsbv 1963 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 1962 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 GG, 25-Aug-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
25-Aug-2024cbvexvw 1932 Change bound variable. See cbvexv 1930 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1459. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
25-Aug-2024cbvalvw 1931 Change bound variable. See cbvalv 1929 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1459. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
25-Aug-2024nfal 1587 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 1521. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦𝜑
 
24-Aug-2024gcdcomd 12114 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 11975 Deduction form of dvds2add 11971. (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 11751 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 15367 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 3610, (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 15368 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
 
16-Aug-2024fprodunsn 11750 Multiply in an additional term in a finite product. See also fprodsplitsn 11779 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 15371 Alternate proof of bj-charfundc 15370. It was expected to be much shorter since it uses bj-charfun 15369 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 15369 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 15368 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 852, which translate to 1o and ∅ respectively by iftrue 3563 and iffalse 3566, giving pwtrufal 15558).

As proved in if0ab 15367, 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 8666 Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 832. 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 8665 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 4516 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 4515 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 4514 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 4513 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 3573 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 3572 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 7284 Lemma for exmidontriim 7287. (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 7283 Lemma for exmidontriim 7287. A variation of r19.30dc 2641. (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 852 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 851 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 1660 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 4225): then, we can prove ∀𝑥 ∈ 𝒫 1o¬ ¬ DECID 𝑥 = 1o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o because the converse of nnral 2484 does not hold.

Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 694). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)

¬ ¬ DECID 𝜑
 
10-Aug-2024exmidontriim 7287 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 7286 Lemma for exmidontriim 7287. 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 7285 Lemma for exmidontriim 7287. 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 7188 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 7185 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 4707 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 6971 Reformulation of ss1o0el1 4227 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 6969 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 4227 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 6972 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 6968 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 4638 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 4639. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ ω → 𝐴 ⊆ ω)
 
6-Aug-2024bj-charfunbi 15373 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 15372 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 15370 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 11735 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 15366 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 15365 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5292, then prove funmptd 15365 from it, and then prove funmpt 5293 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 15317 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊥
 
5-Aug-2024bj-dctru 15315 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊤
 
5-Aug-2024bj-stfal 15304 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊥
 
5-Aug-2024bj-sttru 15302 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊤
 
5-Aug-2024prod1dc 11732 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 6579 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
2-Aug-2024onntri52 7306 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 7304 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 7303 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 7302 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 7300 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 7299 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 7300), (3) implies (5) (onntri35 7299), (5) implies (1) (onntri51 7302), (2) implies (4) (onntri24 7304), (4) implies (5) (onntri45 7303), and (5) implies (2) (onntri52 7306).

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 7301 and exmidontri2or 7305, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7307 and (4) by onntri2or 7308.

(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 2484 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1660. (Contributed by Jim Kingdon, 1-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
 
31-Jul-20243nsssucpw1 7298 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 7297 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-2024psrbagf 14167 A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
 
30-Jul-20243nelsucpw1 7296 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 7295 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 7294 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 7293 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 7292 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 7291 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o
 
30-Jul-2024pw1ne0 7290 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅
 
29-Jul-2024grpcld 13089 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 7288 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
28-Jul-2024exmidpweq 6967 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 15622 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 15621 by means of dceqnconst 15620. (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 15618 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 15624 Lemma for nconstwlpo 15626. 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 15623 Lemma for nconstwlpo 15626. 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 15616 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 15603 and redcwlpo 15615). Thus, this is an analytic analogue to lpowlpo 7229. (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 15611 Weak omniscience stated in terms of equality with 0. Like iswomninn 15610 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 7229 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7228. There is an analogue in terms of analytic omniscience principles at tridceq 15616. (Contributed by Jim Kingdon, 24-Jul-2024.)
(ω ∈ Omni → ω ∈ WOmni)
 
23-Jul-2024nconstwlpolem 15625 Lemma for nconstwlpo 15626. (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 15620 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 15615 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 15617 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 5872 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 1511 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 15626 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 11729 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 2499 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 2498 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 15151 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 12321, 2logb9irrap 15150 and sqrt2cxp2logb9e3 15148. 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 15150 Example for logbgcd1irrap 15143. 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)
 
12-Jul-2024erlecpbl 12918 Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))
 
12-Jul-2024ercpbl 12917 Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)    &   (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
 
12-Jul-2024ercpbllemg 12916 Lemma for ercpbl 12917. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))
 
12-Jul-2024divsfvalg 12915 Value of the function in qusval 12909. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
 
12-Jul-2024divsfval 12914 Value of the function in qusval 12909. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    ⇒   (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
 
11-Jul-2024logbgcd1irraplemexp 15141 Lemma for logbgcd1irrap 15143. 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 14954 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 15113 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 15142 Lemma for logbgcd1irrap 15143. 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 10792 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 15053 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 15118 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 15054 Deduction form of logrpap0 15053. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → (log‘𝐴) # 0)
 
3-Jul-2024logrpap0b 15052 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 15557 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 15556 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 15595 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
27-Jun-2024iooref1o 15594 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 15627 Lemma for neapmkv 15628. 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 15613 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 15612 Lemma for ismkvnn 15613. 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 7222 Lemma for enmkv 7223. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
 
24-Jun-2024neapmkv 15628 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 15621 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 15603 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 15620 and in fact this theorem can be proved using dceqnconst 15620 as shown at dcapnconstALT 15622. (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 7223 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 6485 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 15614 Lemma for redcwlpo 15615. A biconditionalized version of trilpolemeq1 15600. (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 15615 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 15614). 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 10317 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
20-Jun-2024iswomninn 15610 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7227 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 15609 Lemma for iswomnimap 7227. 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 7231 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 6485 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 7230 Lemma for enwomni 7231. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 15114 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 15097 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))
 
16-Jun-2024biadanid 614 Deduction associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
((𝜑 ∧ 𝜓) → 𝜒)    &   ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))    ⇒   (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
 
13-Jun-2024rpcxpadd 15081 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 15080 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 15078 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ)
 
12-Jun-2024rpcxp0 15074 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 15072 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 15071 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 15070 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 15035 Define the power function on complex numbers. Because df-relog 15034 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 15605 Version of trirec0 15604 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 15604 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 15603). (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 7228 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 7227 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 7226 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 7225 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-2024ringcmnd 13534 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ CMnd)
 
1-Jun-2024ringabld 13533 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Abel)
 
1-Jun-2024cmnmndd 13381 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ CMnd)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024ablcmnd 13365 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Abel)    ⇒   (𝜑 → 𝐺 ∈ CMnd)
 
1-Jun-2024grpmndd 13088 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
29-May-2024pw1nct 15563 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 15562 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 4188 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 14903 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 2716 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2718 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 1518 and ax-bndl 1520 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by GG, 23-May-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
22-May-2024efltlemlt 14950 Lemma for eflt 14951. The converse of efltim 11844 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 14951 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 15607 Lemma for apdiff 15608. (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 15606 Lemma for apdiff 15608. 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 15608 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)))))
 
16-May-2024lmodgrpd 13796 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑊 ∈ LMod)    ⇒   (𝜑 → 𝑊 ∈ Grp)
 
16-May-2024crnggrpd 13509 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
16-May-2024crngringd 13508 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
16-May-2024ringgrpd 13504 A ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
15-May-2024reeff1oleme 14948 Lemma for reeff1o 14949. (Contributed by Jim Kingdon, 15-May-2024.)
(𝑈 ∈ (0(,)e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
14-May-2024df-relog 15034 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)
 
14-May-2024fvmpopr2d 6056 Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)
(𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))    &   (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)    &   ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)
 
12-May-2024dvdstrd 11976 The divides relation is transitive, a deduction version of dvdstr 11974. (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 15030 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 15028 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 15029 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 12604 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12600 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 12601 Variation of ctiunct 12600 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 7333 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7332, 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 7331 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 7332 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 7330 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 7329 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 7328 Lemma for cc2 7329. (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 7327 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))
 
24-Apr-2024lsppropd 13931 If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   (𝜑 → 𝐵 ⊆ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))    &   (𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝐿 ∈ 𝑌)    ⇒   (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
 
19-Apr-2024omctfn 12603 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 11723 Lemma for prodmodc 11724. (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 11722 Lemma for prodmodc 11724. (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 11721 Lemma for prodmodc 11724. (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 653 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 11717 Lemma for prodrbdc 11720. (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 11693 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 11692 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 11691 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 11687 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 11697 Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sumdc 11500 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 15027 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 15026 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 15010 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 15009 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 14959 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)
 
9-Mar-2024exmidonfin 7256 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 6930 and nnon 4643. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(ω = (On ∩ Fin) → EXMID)
 
9-Mar-2024exmidonfinlem 7255 Lemma for exmidonfin 7256. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   (ω = (On ∩ Fin) → DECID 𝜑)
 
8-Mar-2024sin0pilem2 14958 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 14957 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 14956 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 11914 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-2024scaffvalg 13805 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    ∙ = ( ·sf ‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    ⇒   (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
 
2-Mar-2024dvrfvald 13632 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → · = (.r‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → 𝐼 = (invr‘𝑅))    &   (𝜑 → / = (/r‘𝑅))    &   (𝜑 → 𝑅 ∈ SRing)    ⇒   (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
 
2-Mar-2024plusffvalg 12948 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 13060 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 9833 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 9832 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 8669 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 8652 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 8670 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 14823 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 14821 Lemma for ivthinc 14822. 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 14817 Lemma for ivthinc 14822. 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 14812 A Dedekind cut identifies a unique real number. Similar to df-inp 7528 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 13120 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 14820 Lemma for ivthinc 14822. 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 14819 Lemma for ivthinc 14822. 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 14818 Lemma for ivthinc 14822. 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 14816 Lemma for ivthinc 14822. 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 14814 Lemma for ivthinc 14822. 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 14813 Lemma for ivthinc 14822. 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 13059 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
17-Feb-2024mndissubm 13050 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 12947 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 14810 Lemma for dedekindicc 14812. 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 14809 Lemma for dedekindicc 14812. 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 14808 Lemma for dedekindicc 14812. 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 14807 Lemma for dedekindicc 14812. 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 14806 Lemma for dedekindicc 14812. 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 14805 Lemma for dedekindicc 14812. 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 14804 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8094 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 14803 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8094 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)))
 
10-Feb-2024cbvexdvaw 1943 Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1941 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Feb-2024cbvaldvaw 1942 Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1940 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
6-Feb-2024ivthinclemlopn 14815 Lemma for ivthinc 14822. 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 14822 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 14796 Lemma for dedekindeu 14802. 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 14801 Lemma for dedekindeu 14802. 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 14800 Lemma for dedekindeu 14802. 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 14799 Lemma for dedekindeu 14802. 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 14798 Lemma for dedekindeu 14802. 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 14797 Lemma for dedekindeu 14802. 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 8094 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 7995 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 5247 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 13004 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)
 
26-Jan-2024elovmporab1w 6121 Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by GG, 26-Jan-2024.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
 
26-Jan-2024opabidw 4288 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4287 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
 
24-Jan-2024axpre-suploclemres 7963 Lemma for axpre-suploc 7964. 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 7871. (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 7995 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 7994 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7994.

(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 7964 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 7995. (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 7871 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 15336 Shorter proof of el2oss1o 6498 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o → 𝐴 ⊆ 1o)
 
21-Jan-2024ltm1sr 7839 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 13029 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 7869 Lemma for suplocsr 7871. 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 7868 Lemma for suplocsr 7871. 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 7870 Lemma for suplocsr 7871. 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)))
 
15-Jan-2024eqg0el 13302 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
∼ = (𝐺 ~QG 𝐻)    ⇒   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))
 
14-Jan-2024suplocexprlemlub 7786 Lemma for suplocexpr 7787. 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 7785 Lemma for suplocexpr 7787. 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 3118 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3119 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
 
10-Jan-2024nfsbcw 3116 Bound-variable hypothesis builder for class substitution. Version of nfsbc 3007 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥[𝐴 / 𝑦]𝜑
 
10-Jan-2024nfsbcdw 3115 Version of nfsbcd 3006 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvcsbw 3085 Version of cbvcsb 3086 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
Ⅎ𝑦𝐶    &   Ⅎ𝑥𝐷    &   (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
 
10-Jan-2024cbvsbcw 3014 Version of cbvsbc 3015 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvrex2vw 2738 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2740 with a disjoint variable condition, which does not require ax-13 2166. (Contributed by FL, 2-Jul-2012.) (Revised by GG, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvral2vw 2737 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2739 with a disjoint variable condition, which does not require ax-13 2166. (Contributed by NM, 10-Aug-2004.) (Revised by GG, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvrexw 2721 Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2717 with more disjoint variable conditions. Although we don't do so yet, we expect the disjoint variable conditions will allow us to remove reliance on ax-i12 1518 and ax-bndl 1520 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvralw 2720 Rule used to change bound variables, using implicit substitution. Version of cbvral 2722 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 1518 and ax-bndl 1520 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexfw 2717 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2719 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 1518 and ax-bndl 1520 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfralw 2531 Bound-variable hypothesis builder for restricted quantification. See nfralya 2534 for a version with 𝑦 and 𝐴 distinct instead of 𝑥 and 𝑦. (Contributed by NM, 1-Sep-1999.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfraldw 2526 Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2529 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfabdw 2355 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2356 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
10-Jan-2024cbvex2vw 1945 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) (Revised by GG, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
 
10-Jan-2024cbval2vw 1944 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
 
10-Jan-2024cbv2w 1761 Rule used to change bound variables, using implicit substitution. Version of cbv2 1760 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
9-Jan-2024suplocexprlemloc 7783 Lemma for suplocexpr 7787. 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 7782 Lemma for suplocexpr 7787. 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 7781 Lemma for suplocexpr 7787. 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 7779 Lemma for suplocexpr 7787. 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 7776 Lemma for suplocexpr 7787. 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 7775 Lemma for suplocexpr 7787. 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 7787 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 7784 Lemma for suplocexpr 7787. 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 7780 Lemma for suplocexpr 7787. 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 7778 Lemma for suplocexpr 7787. 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 7777 Lemma for suplocexpr 7787. 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 14811 Lemma for dedekindicc 14812. Same as dedekindicc 14812, 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 14802 A Dedekind cut identifies a unique real number. Similar to df-inp 7528 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 14902 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 14901 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 14900 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-2023rinvmod 13382 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6114. (Contributed by AV, 31-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )
 
31-Dec-2023brm 4080 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 14894 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 14893 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)
 
30-Dec-2023eqwrd 10957 Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
((𝑈 ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))
 
29-Dec-2023mndbn0 13015 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 13014). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
28-Dec-2023mulgnn0gsum 13201 Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023mulgnngsum 13200 Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
26-Dec-2023gsumfzreidx 13410 Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
 
26-Dec-2023gsumsplit1r 12984 Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
26-Dec-2023lidrididd 12968 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 12967) 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 12967 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 7179 A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)
 
23-Dec-2023enct 12593 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 12592 Lemma for enct 12593. 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 7178 ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
 
21-Dec-2023dvcoapbr 14886 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 8668 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 8667 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
 
18-Dec-2023limccoap 14857 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 11125 Complex apartness in terms of real and imaginary parts. See also apreim 8624 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 14790 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 14375 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 14725 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 14723 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 14721 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 15312 Clavius law for stable formulas. See pm2.18dc 856. (Contributed by BJ, 4-Dec-2023.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
 
4-Dec-2023bj-nnclavius 15299 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
 
2-Dec-2023dvmulxx 14883 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14881. (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 14881 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14883. (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 15561 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 7216 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-2023ccfunen 7326 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))
 
28-Nov-2023exmid1stab 4238 If every 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)
 
27-Nov-2023df-cc 7325 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7268 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 6146 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 14882 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14880. (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 14880 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 849 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 843. The relation between dcn 843 and dcnn 849 is analogous to that between notnot 630 and notnotnot 635 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 849 means that a proposition is testable if and only if its negation is testable, and dcn 843 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 15323 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 15319 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 15306. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-dcstab 15318 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-fadc 15316 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → DECID 𝜑)
 
24-Nov-2023bj-trdc 15314 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → DECID 𝜑)
 
24-Nov-2023bj-stal 15311 The universal quantification of a stable formula is stable. See bj-stim 15308 for implication, stabnot 834 for negation, and bj-stan 15309 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)
 
24-Nov-2023bj-stand 15310 The conjunction of two stable formulas is stable. Deduction form of bj-stan 15309. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 15309 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 15309 The conjunction of two stable formulas is stable. See bj-stim 15308 for implication, stabnot 834 for negation, and bj-stal 15311 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))
 
24-Nov-2023bj-stim 15308 A conjunction with a stable consequent is stable. See stabnot 834 for negation , bj-stan 15309 for conjunction , and bj-stal 15311 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB 𝜓 → STAB (𝜑 → 𝜓))
 
24-Nov-2023bj-nnbist 15306 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 15319). (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-fast 15303 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-trst 15301 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → STAB 𝜑)
 
24-Nov-2023bj-nnan 15298 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 15297 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 15295 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 1660 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 6142 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 7271 The axiom of choice implies excluded middle. See acexmid 5918 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 7270 Lemma for exmidac 7271. 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 4230 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4223 or ordtriexmid 4554. 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 7269 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-2023rnrhmsubrg 13751 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
 
18-Nov-2023condc 854 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 848 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 843. (Contributed by BJ, 18-Nov-2023.)
(STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑))
 
17-Nov-2023cnplimclemr 14848 Lemma for cnplimccntop 14849. 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 14847 Lemma for cnplimccntop 14849. 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 14856 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 11370 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 14855 Lemma for limccnp2cntop 14856. 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 14839 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 14842 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 12602 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 12600 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 12604 (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 12602, 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 12555) 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 7172 and ctssdc 7174.

(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 7172 A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7174 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 12599 Lemma for ctiunct 12600. (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 12598 Lemma for ctiunct 12600. (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 12597 Lemma for ctiunct 12600. (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 12596 Lemma for ctiunct 12600. (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 12595 Lemma for ctiunct 12600. (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 12594 Lemma for ctiunct 12600. (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 869 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 846 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 655 Weakening of conax1 654. General instance of pm2.51 656 and of pm2.52 657. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → (𝜒 → ¬ 𝜓))
 
28-Oct-2023conax1 654 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → ¬ 𝜓)
 
25-Oct-2023divcnap 14744 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 7894 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 6226 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 14740 Lemma for addcncntop 14741, subcncntop 14742, and mulcncntop 14743. (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 14697 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 14687 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 14688 Lemma for xmettx 14689. (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 14689 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 14686 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)))
 
7-Oct-2023df-iress 12629 Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)

↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
 
29-Sep-2023syl2anc2 412 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(𝜑 → 𝜓)    &   (𝜓 → 𝜒)    &   ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   (𝜑 → 𝜃)
 
27-Sep-2023fnpr2ob 12926 Biconditional version of fnpr2o 12925. (Contributed by Jim Kingdon, 27-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
25-Sep-2023xpsval 12938 Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))
 
25-Sep-2023fvpr1o 12928 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐵 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
 
25-Sep-2023fvpr0o 12927 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐴 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
 
25-Sep-2023fnpr2o 12925 Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
25-Sep-2023df-xps 12890 Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
 
12-Sep-2023pwntru 4229 A slight strengthening of pwtrufal 15558. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
((𝐴 ⊆ {∅} ∧ 𝐴 ≠ {∅}) → 𝐴 = ∅)
 
11-Sep-2023pwtrufal 15558 A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4228. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4226), 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 15294 (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 7105 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 15560 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 15559 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 15591 Omniscience stated in terms of natural numbers. Similar to isomnimap 7198 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 15590 Lemma for isomninn 15591. 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 15598 Lemma for trilpo 15603. 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 15593 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 15592 Lemma for cvgcmp2n 15593. 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 15596 Lemma for trilpo 15603. 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 15603 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 15601 (which means the sequence contains a zero), trilpolemeq1 15600 (which means the sequence is all ones), and trilpolemgt1 15599 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15589) or that the real numbers are a discrete field (see trirec0 15604).

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 10313 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 15602 Lemma for trilpo 15603. 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 15601 Lemma for trilpo 15603. 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 7201 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 15600 Lemma for trilpo 15603. 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 15599 Lemma for trilpo 15603. 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 15597 Lemma for trilpo 15603. 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 15589 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 7274 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 7173 Lemma for ctssdc 7174. 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 7171 Lemma for ctssdc 7174. 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 7211 The decidability condition in ctssdc 7174 is needed. More specifically, ctssdc 7174 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 7174 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 7211. (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 6268 Weak version of mpoex 6269 that holds without ax-coll 4145. 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 13117 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 8149 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 6167 Weak version of mptex 5785 that holds without ax-coll 4145. 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 6166 Weak version of funex 5782 that holds without ax-coll 4145. 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 12591 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 12587 Lemma for ctinfom 12588. 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 7160 Lemma for difinfsn 7161. 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 7162 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 7161 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 12590 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 12589 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 12588 A condition for a set being countably infinite. Restates ennnfone 12585 in terms of ω and function image. Like ennnfone 12585 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 14737 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 14736 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 15580 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 14433 Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 14439. (Contributed by Jim Kingdon, 3-Aug-2023.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)
 
3-Aug-2023ablgrpd 13363 An Abelian group is a group, deduction form of ablgrp 13362. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → 𝐺 ∈ Abel)    ⇒   (𝜑 → 𝐺 ∈ Grp)
 
3-Aug-20231nsgtrivd 13292 A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)    ⇒   (𝜑 → 𝐵 = { 0 })
 
3-Aug-2023triv1nsgd 13291 A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = { 0 })    ⇒   (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)
 
3-Aug-2023trivnsgd 13290 The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = { 0 })    ⇒   (𝜑 → (NrmSGrp‘𝐺) = {𝐵})
 
3-Aug-20230idnsgd 13289 The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺))
 
3-Aug-2023trivsubgsnd 13274 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = { 0 })    ⇒   (𝜑 → (SubGrp‘𝐺) = {𝐵})
 
3-Aug-2023trivsubgd 13273 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = { 0 })    &   (𝜑 → 𝐴 ∈ (SubGrp‘𝐺))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
3-Aug-2023mulgcld 13217 Deduction associated with mulgcl 13212. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
 
3-Aug-2023hashfingrpnn 13111 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 13016 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 12134 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 12133 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 10871 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 6991 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6923 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 6873 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 4918 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 4917 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 2872 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 2198 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → ¬ 𝐴 = 𝐵)    ⇒   (𝜑 → ¬ 𝐵 = 𝐴)
 
2-Aug-2023dvid 14874 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 14873 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 14870 Lemma for dvid 14874 and dvconst 14873. (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 8857 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8726. Converse of diveqap1d 8819. (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 11387 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11209 and mulap0bd 8678 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 8691 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 8645 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 10510 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 7273 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 7272 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 7159 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 7158 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 12586 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 4233 Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4228 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4232 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 14869 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 14868 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)
 
24-Jul-2023sraring 13948 Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring)
 
23-Jul-2023ennnfonelemhdmp1 12569 Lemma for ennnfone 12585. 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 12566 Lemma for ennnfone 12585. 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 6558 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 6557 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))
 
22-Jul-2023relopabiv 4786 A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4788. (Contributed by BJ, 22-Jul-2023.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   Rel 𝐴
 
21-Jul-2023ennnfoneleminc 12571 Lemma for ennnfone 12585. 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 12563 Lemma for ennnfone 12585. 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 12562 Lemma for ennnfone 12585. 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 12561 Lemma for ennnfone 12585. 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 10544 Value of the sequence builder function at a successor. A version of seq3p1 10539 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 10541 A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10542 and seq1cd 10543 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 12575 Lemma for ennnfone 12585. 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 12574 Lemma for ennnfone 12585. 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 12572 Lemma for ennnfone 12585. 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 12568 Lemma for ennnfone 12585. 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 12567 Lemma for ennnfone 12585. 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 10543 Initial value of the recursive sequence builder. A version of seq3-1 10536 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 12573 Lemma for ennnfone 12585. 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 12581 Lemma for ennnfone 12585. 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 12580 Lemma for ennnfone 12585. 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 12579 Lemma for ennnfone 12585. 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 12578 Lemma for ennnfone 12585. 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 12577 Lemma for ennnfone 12585. 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 12576 Lemma for ennnfone 12585. A consequence of ennnfonelemss 12570. (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 12570 Lemma for ennnfone 12585. 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 12565 Lemma for ennnfone 12585. 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 12560 Lemma for ennnfone 12585. (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 12559 Lemma for ennnfone 12585. A direct consequence of fidcenumlemrk 7015. (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 7130 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 15585 Lemma for sbthom 15586. (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 7153 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 7126 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 7282 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 7281 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 7155 Lemma for omp1eom 7156. (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 6489 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 15586 Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 15584 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 7157 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 7156 Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
(ω ⊔ 1o) ≈ ω
 
9-Jul-2023refeq 15588 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 10535 Value of the sequence builder function. Similar to seq3val 10534 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 7128 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 7125 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 14844 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 12564 Lemma for ennnfone 12585. (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 10542 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)
 
6-Jul-2023sbbidv 1896 Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 1857. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑥]𝜒))
 
3-Jul-2023limcimolemlt 14843 Lemma for limcimo 14844. (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 14867 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 14865 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 14863 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 14861 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 14860 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 14859 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 14858 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 14836 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 14854 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 2641 Restricted quantifier version of 19.30dc 1638. (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 2622 Restricted quantifier version of one direction of 19.28 1574. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3540.) (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 2621 Restricted quantitifer version of one direction of 19.27 1572. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3544.) (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 14850 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 14773 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 14846 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 14699 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 14712 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   𝐽 ∈ Top
 
6-Jun-2023cntoptopon 14711 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 14841 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 14840 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 14835 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 12388 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 11946 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 9636 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 9635 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 14786 Lemma for mulcncf 14787. (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 14783 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 11459 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 2193. (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 14707 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 11383 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 9364 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 14700 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 11423 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 11500 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 11668). (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 14683 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 14682 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 14681 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 11418 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 2898 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 11419 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 2633 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 11385 Lemma for bdtri 11386. (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 11422 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 11386 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 11382 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))
 
13-May-2023kerf1ghm 13347 A group homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g‘𝑅)    &    0 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))
 
13-May-2023f1ghm0to0 13345 If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g‘𝑅)    &    0 = (0g‘𝑆)    ⇒   ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁))
 
11-May-2023xrmaxadd 11407 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 11406 Lemma for xrmaxadd 11407. 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 11421 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 11405 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 11403 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 14680 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 14679 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 14660 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 12833 Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
(dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ)
 
6-May-2023eqabdv 2322 Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Revised by Wolf Lammen, 6-May-2023.)
(𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))    ⇒   (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
 
5-May-2023mopnrel 14620 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
Rel MetOpen
 
5-May-2023fsumsersdc 11541 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 14566 A ball is a set. Also see blfn 14050 in case you just know 𝐷 is a set, not 𝐷 ∈ (∞Met‘𝑋). (Contributed by Jim Kingdon, 4-May-2023.)
(𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V)
 
4-May-2023summodc 11529 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 11528 Lemma for summodc 11529. (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 11420 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 11417 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 11416 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)))
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