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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | df3nandALT2 36401 | The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
| Theorem | andnand1 36402 | Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
| ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ⊼ (𝜑 ⊼ 𝜓 ⊼ 𝜒))) | ||
| Theorem | imnand2 36403 | An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
| ⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) | ||
| Theorem | nalfal 36404 | Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∀𝑥⊥ | ||
| Theorem | nexntru 36405 | There does not exist a set such that ⊤ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∃𝑥 ¬ ⊤ | ||
| Theorem | nexfal 36406 | There does not exist a set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∃𝑥⊥ | ||
| Theorem | neufal 36407 | There does not exist exactly one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∃!𝑥⊥ | ||
| Theorem | neutru 36408 | There does not exist exactly one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∃!𝑥⊤ | ||
| Theorem | nmotru 36409 | There does not exist at most one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ¬ ∃*𝑥⊤ | ||
| Theorem | mofal 36410 | There exist at most one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ ∃*𝑥⊥ | ||
| Theorem | nrmo 36411 | "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
| ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑) ⇒ ⊢ ∃*𝑥 ∈ 𝐴 𝜑 | ||
| Theorem | meran1 36412 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
| ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑)))) | ||
| Theorem | meran2 36413 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
| ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃)))) | ||
| Theorem | meran3 36414 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
| ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑)))) | ||
| Theorem | waj-ax 36415 | A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.) |
| ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓)))) | ||
| Theorem | lukshef-ax2 36416 | A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.) |
| ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
| Theorem | arg-ax 36417 | A single axiom for propositional calculus discovered by Ken Harris and Branden Fitelson. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom HF1 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.) |
| ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
| Theorem | negsym1 36418 |
In the paper "On Variable Functors of Propositional Arguments",
Lukasiewicz introduced a system that can handle variable connectives.
This was done by introducing a variable, marked with a lowercase delta,
which takes a wff as input. In the system, "delta 𝜑 "
means that
"something is true of 𝜑". The expression "delta
𝜑
" can be
substituted with ¬ 𝜑, 𝜓 ∧ 𝜑, ∀𝑥𝜑, etc.
Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus. A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ (¬ ¬ ⊥ → ¬ 𝜑) | ||
| Theorem | imsym1 36419 |
A symmetry with →.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑)) | ||
| Theorem | bisym1 36420 |
A symmetry with ↔.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑)) | ||
| Theorem | consym1 36421 |
A symmetry with ∧.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)) | ||
| Theorem | dissym1 36422 |
A symmetry with ∨.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑)) | ||
| Theorem | nandsym1 36423 |
A symmetry with ⊼.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) | ||
| Theorem | unisym1 36424 |
A symmetry with ∀.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
| ⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑) | ||
| Theorem | exisym1 36425 |
A symmetry with ∃.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
| ⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑) | ||
| Theorem | unqsym1 36426 |
A symmetry with ∃!.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) |
| ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | ||
| Theorem | amosym1 36427 |
A symmetry with ∃*.
See negsym1 36418 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
| ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) | ||
| Theorem | subsym1 36428 |
A symmetry with [𝑥 / 𝑦].
See negsym1 36418 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
| ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑) | ||
| Theorem | ontopbas 36429 | An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
| ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) | ||
| Theorem | onsstopbas 36430 | The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.) |
| ⊢ On ⊆ TopBases | ||
| Theorem | onpsstopbas 36431 | The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.) |
| ⊢ On ⊊ TopBases | ||
| Theorem | ontgval 36432 | The topology generated from an ordinal number 𝐵 is suc ∪ 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.) |
| ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵) | ||
| Theorem | ontgsucval 36433 | The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
| ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | ||
| Theorem | onsuctop 36434 | A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
| ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) | ||
| Theorem | onsuctopon 36435 | One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
| ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) | ||
| Theorem | ordtoplem 36436 | Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
| ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) ⇒ ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) | ||
| Theorem | ordtop 36437 | An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.) |
| ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | ||
| Theorem | onsucconni 36438 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
| ⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ Conn | ||
| Theorem | onsucconn 36439 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
| ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) | ||
| Theorem | ordtopconn 36440 | An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
| ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) | ||
| Theorem | onintopssconn 36441 | An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
| ⊢ (On ∩ Top) ⊆ Conn | ||
| Theorem | onsuct0 36442 | A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.) |
| ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2) | ||
| Theorem | ordtopt0 36443 | An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.) |
| ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2)) | ||
| Theorem | onsucsuccmpi 36444 | The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.) |
| ⊢ 𝐴 ∈ On ⇒ ⊢ suc suc 𝐴 ∈ Comp | ||
| Theorem | onsucsuccmp 36445 | The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.) |
| ⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) | ||
| Theorem | limsucncmpi 36446 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
| ⊢ Lim 𝐴 ⇒ ⊢ ¬ suc 𝐴 ∈ Comp | ||
| Theorem | limsucncmp 36447 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
| ⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) | ||
| Theorem | ordcmp 36448 | An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1o. (Contributed by Chen-Pang He, 1-Nov-2015.) |
| ⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o))) | ||
| Theorem | ssoninhaus 36449 | The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.) |
| ⊢ {1o, 2o} ⊆ (On ∩ Haus) | ||
| Theorem | onint1 36450 | The ordinal T1 spaces are 1o and 2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.) |
| ⊢ (On ∩ Fre) = {1o, 2o} | ||
| Theorem | oninhaus 36451 | The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.) |
| ⊢ (On ∩ Haus) = {1o, 2o} | ||
| Theorem | fveleq 36452 | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
| ⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃))) | ||
| Theorem | findfvcl 36453* | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
| ⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃) & ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃))) ⇒ ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) | ||
| Theorem | findreccl 36454* | Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.) |
| ⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)) | ||
| Theorem | findabrcl 36455* | Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.) |
| ⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃) | ||
| Theorem | nnssi2 36456 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
| ⊢ ℕ ⊆ 𝐷 & ⊢ (𝐵 ∈ ℕ → 𝜑) & ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓) | ||
| Theorem | nnssi3 36457 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
| ⊢ ℕ ⊆ 𝐷 & ⊢ (𝐶 ∈ ℕ → 𝜑) & ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) | ||
| Theorem | nndivsub 36458 | Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
| ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝐶) ∈ ℕ)) | ||
| Theorem | nndivlub 36459 | A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
| ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵 ≤ 𝐴)) | ||
| Syntax | cgcdOLD 36460 | Extend class notation to include the gdc function. (New usage is discouraged.) |
| class gcdOLD (𝐴, 𝐵) | ||
| Definition | df-gcdOLD 36461* | gcdOLD (𝐴, 𝐵) is the largest positive integer that evenly divides both 𝐴 and 𝐵. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.) |
| ⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < ) | ||
| Theorem | ee7.2aOLD 36462 | Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴)))) | ||
| Theorem | weiunlem1 36463* | Lemma for weiunpo 36466, weiunso 36467, weiunfr 36468, and weiunse 36469. (Contributed by Matthew House, 8-Sep-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ (𝐶𝑇𝐷 ↔ ((𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐷 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷)))) | ||
| Theorem | weiunlem2 36464* | Lemma for weiunpo 36466, weiunso 36467, weiunfr 36468, and weiunse 36469. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡))) | ||
| Theorem | weiunfrlem 36465* | Lemma for weiunfr 36468. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ 𝐸 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) & ⊢ (𝜑 → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) & ⊢ (𝜑 → 𝑟 ≠ ∅) ⇒ ⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸)) | ||
| Theorem | weiunpo 36466* | A partial ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of partial orderings on its members. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Po 𝐵) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | weiunso 36467* | A strict ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of strict orderings on its members. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | weiunfr 36468* | A well-founded relation on an indexed union can be constructed from a well-ordering on its index class and a collection of well-founded relations on its members. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | weiunse 36469* | The relation constructed in weiunpo 36466, weiunso 36467, weiunfr 36468, and weiunwe 36470 is set-like if all members of the indexed union are sets. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → 𝑇 Se ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | weiunwe 36470* | A well-ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) & ⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | numiunnum 36471* | An indexed union of sets is numerable if its index set is numerable and there exists a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.) |
| ⊢ ((𝐴 ∈ dom card ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card) | ||
| Theorem | dnival 36472* | Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) | ||
| Theorem | dnicld1 36473 | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) | ||
| Theorem | dnicld2 36474* | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) | ||
| Theorem | dnif 36475 | The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ 𝑇:ℝ⟶ℝ | ||
| Theorem | dnizeq0 36476* | The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) = 0) | ||
| Theorem | dnizphlfeqhlf 36477* | The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) | ||
| Theorem | rddif2 36478 | Variant of rddif 15379. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) | ||
| Theorem | dnibndlem1 36479* | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆)) | ||
| Theorem | dnibndlem2 36480* | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | dnibndlem3 36481 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1)) ⇒ ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (abs‘((𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) + (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)))) | ||
| Theorem | dnibndlem4 36482 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
| Theorem | dnibndlem5 36483 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) | ||
| Theorem | dnibndlem6 36484 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) + ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) | ||
| Theorem | dnibndlem7 36485 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
| Theorem | dnibndlem8 36486 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) | ||
| Theorem | dnibndlem9 36487* | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1)) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | dnibndlem10 36488 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴)) | ||
| Theorem | dnibndlem11 36489 | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2)) | ||
| Theorem | dnibndlem12 36490* | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | dnibndlem13 36491* | Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | dnibnd 36492* | The "distance to nearest integer" function is 1-Lipschitz continuous, i.e., is a short map. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | dnicn 36493 | The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ 𝑇 ∈ (ℝ–cn→ℝ) | ||
| Theorem | knoppcnlem1 36494* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
| ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) | ||
| Theorem | knoppcnlem2 36495* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ) | ||
| Theorem | knoppcnlem3 36496* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) ∈ ℝ) | ||
| Theorem | knoppcnlem4 36497* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (abs‘((𝐹‘𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀)) | ||
| Theorem | knoppcnlem5 36498* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑m ℝ)) | ||
| Theorem | knoppcnlem6 36499* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ)) | ||
| Theorem | knoppcnlem7 36500* | Lemma for knoppcn 36505. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑤))‘𝑀))) | ||
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