HomeTheorem List (p. 447 of 503)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 447 of 503) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-30989) |
Hilbert Space Explorer
(30990-32512) |
Users' Mathboxes
(32513-50274) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | int-addassocd 44601 | AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) | ||
| Theorem | int-addsimpd 44602 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 = (𝐴 − 𝐵)) | ||
| Theorem | int-mulcomd 44603 | MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) | ||
| Theorem | int-mulassocd 44604 | MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷)) | ||
| Theorem | int-mulsimpd 44605 | MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → 1 = (𝐴 / 𝐵)) | ||
| Theorem | int-leftdistd 44606 | AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) | ||
| Theorem | int-rightdistd 44607 | AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) | ||
| Theorem | int-sqdefd 44608 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2)) | ||
| Theorem | int-mul11d 44609 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐵) | ||
| Theorem | int-mul12d 44610 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐵) | ||
| Theorem | int-add01d 44611 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐵) | ||
| Theorem | int-add02d 44612 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐵) | ||
| Theorem | int-sqgeq0d 44613 | SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | ||
| Theorem | int-eqprincd 44614 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷)) | ||
| Theorem | int-eqtransd 44615 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
| Theorem | int-eqmvtd 44616 | EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → 𝐶 = (𝐵 − 𝐷)) | ||
| Theorem | int-eqineqd 44617 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) | ||
| Theorem | int-ineqmvtd 44618 | IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → (𝐵 − 𝐷) ≤ 𝐶) | ||
| Theorem | int-ineq1stprincd 44619 | FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐷 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐵 + 𝐷) ≤ (𝐴 + 𝐶)) | ||
| Theorem | int-ineq2ndprincd 44620 | SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐵 · 𝐶) ≤ (𝐴 · 𝐶)) | ||
| Theorem | int-ineqtransd 44621 | InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐴) | ||
This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12659 addcomli 11338 00id 11321 addridi 11333 addlidi 11334 eqid 2737 dec0h 12666 decadd 12698 decaddc 12699. | ||
| Theorem | unitadd 44622 | Theorem used in conjunction with decaddc 12699 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝐴 + 𝐵) = 𝐹 & ⊢ (𝐶 + 1) = 𝐵 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ ((𝐴 + 𝐶) + 1) = 𝐹 | ||
| Theorem | gsumws3 44623 | Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ 𝑈 ∈ 𝐵))) → (𝐺 Σg ⟨“𝑆𝑇𝑈”⟩) = (𝑆 + (𝑇 + 𝑈))) | ||
| Theorem | gsumws4 44624 | Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg ⟨“𝑆𝑇𝑈𝑉”⟩) = (𝑆 + (𝑇 + (𝑈 + 𝑉)))) | ||
| Theorem | amgm2d 44625 | Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 26953. (Contributed by Stanislas Polu, 8-Sep-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) | ||
| Theorem | amgm3d 44626 | Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3)) | ||
| Theorem | amgm4d 44627 | Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) | ||
| Theorem | spALT 44628 | sp 2191 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2191 instead. (New usage is discouraged.) |
| ⊢ (∀𝑥𝜑 → 𝜑) | ||
| Theorem | elnelneqd 44629 | Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
| Theorem | elnelneq2d 44630 | Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
| Theorem | rr-spce 44631* | Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
| ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
| Theorem | rexlimdvaacbv 44632* | Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3140. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
| Theorem | rexlimddvcbvw 44633* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44632. The equivalent of this theorem without the bound variable change is rexlimddv 3145. Version of rexlimddvcbv 44634 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 2-Apr-2024.) |
| ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | rexlimddvcbv 44634* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44632. The equivalent of this theorem without the bound variable change is rexlimddv 3145. Usage of this theorem is discouraged because it depends on ax-13 2377, see rexlimddvcbvw 44633 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) |
| ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | rr-elrnmpt3d 44635* | Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) | ||
| Theorem | rr-phpd 44636 | Equivalent of php 9141 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | tfindsd 44637* | Deduction associated with tfinds 7811. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
| ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = suc 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝑦 ∈ On ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝜃) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → 𝜂) | ||
| Syntax | cmnring 44638 | Extend class notation with the monoid ring function. |
| class MndRing | ||
| Definition | df-mnring 44639* | Define the monoid ring function. This takes a monoid 𝑀 and a ring 𝑅 and produces a free left module over 𝑅 with a product extending the monoid function on 𝑀. (Contributed by Rohan Ridenour, 13-May-2024.) |
| ⊢ MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩)) | ||
| Theorem | mnringvald 44640* | Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩)) | ||
| Theorem | mnringnmulrd 44641 | Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (.r‘ndx) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) | ||
| Theorem | mnringbased 44642 | The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) | ||
| Theorem | mnringbaserd 44643 | The base set of a monoid ring. Converse of mnringbased 44642. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑉)) | ||
| Theorem | mnringelbased 44644 | Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝐶 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑m 𝐴) ∧ 𝑋 finSupp 0 ))) | ||
| Theorem | mnringbasefd 44645 | Elements of a monoid ring are functions. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝐶 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋:𝐴⟶𝐶) | ||
| Theorem | mnringbasefsuppd 44646 | Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 finSupp 0 ) | ||
| Theorem | mnringaddgd 44647 | The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹)) | ||
| Theorem | mnring0gd 44648 | The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (0g‘𝑉) = (0g‘𝐹)) | ||
| Theorem | mnring0g2d 44649 | The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 × { 0 }) = (0g‘𝐹)) | ||
| Theorem | mnringmulrd 44650* | The ring product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))) = (.r‘𝐹)) | ||
| Theorem | mnringscad 44651 | The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) | ||
| Theorem | mnringvscad 44652 | The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹)) | ||
| Theorem | mnringlmodd 44653 | Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝐹 ∈ LMod) | ||
| Theorem | mnringmulrvald 44654* | Value of multiplication in a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ ∙ = (.r‘𝑅) & ⊢ 𝟎 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ · = (.r‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 ))))) | ||
| Theorem | mnringmulrcld 44655 | Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.) |
| ⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ · = (.r‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) | ||
| Theorem | gru0eld 44656 | A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐴 ∈ 𝐺) ⇒ ⊢ (𝜑 → ∅ ∈ 𝐺) | ||
| Theorem | grusucd 44657 | Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐴 ∈ 𝐺) ⇒ ⊢ (𝜑 → suc 𝐴 ∈ 𝐺) | ||
| Theorem | r1rankcld 44658 | Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ (𝑅1‘𝑅)) ⇒ ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘𝑅)) | ||
| Theorem | grur1cld 44659 | Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐴 ∈ 𝐺) ⇒ ⊢ (𝜑 → (𝑅1‘𝐴) ∈ 𝐺) | ||
| Theorem | grurankcld 44660 | Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐴 ∈ 𝐺) ⇒ ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) | ||
| Theorem | grurankrcld 44661 | If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐺) | ||
| Syntax | cscott 44662 | Extend class notation with the Scott's trick operation. |
| class Scott 𝐴 | ||
| Definition | df-scott 44663* | Define the Scott operation. This operation constructs a subset of the input class which is nonempty whenever its input is using Scott's trick. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} | ||
| Theorem | scotteqd 44664 | Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | scotteq 44665 | Closed form of scotteqd 44664. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | nfscott 44666 | Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Scott 𝐴 | ||
| Theorem | scottabf 44667* | Value of the Scott operation at a class abstraction. Variant of scottab 44668 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))} | ||
| Theorem | scottab 44668* | Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))} | ||
| Theorem | scottabes 44669* | Value of the Scott operation at a class abstraction. Variant of scottab 44668 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
| ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} | ||
| Theorem | scottss 44670 | Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Scott 𝐴 ⊆ 𝐴 | ||
| Theorem | elscottab 44671* | An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓) | ||
| Theorem | scottex2 44672 | scottex 9809 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ Scott 𝐴 ∈ V | ||
| Theorem | scotteld 44673* | The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴) | ||
| Theorem | scottelrankd 44674 | Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) & ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴) ⇒ ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶)) | ||
| Theorem | scottrankd 44675 | Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) ⇒ ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵)) | ||
| Theorem | gruscottcld 44676 | If a Grothendieck universe contains an element of a Scott's trick set, it contains the Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐵 ∈ 𝐺) & ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) ⇒ ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺) | ||
| Syntax | ccoll 44677 | Extend class notation with the collection operation. |
| class (𝐹 Coll 𝐴) | ||
| Definition | df-coll 44678* | Define the collection operation. This is similar to the image set operation “, but it uses Scott's trick to ensure the output is always a set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) | ||
| Theorem | dfcoll2 44679* | Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} | ||
| Theorem | colleq12d 44680 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵)) | ||
| Theorem | colleq1 44681 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴)) | ||
| Theorem | colleq2 44682 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵)) | ||
| Theorem | nfcoll 44683 | Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹 Coll 𝐴) | ||
| Theorem | collexd 44684 | The output of the collection operation is a set if the second input is. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ V) | ||
| Theorem | cpcolld 44685* | Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝑥𝐹𝑦) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | ||
| Theorem | cpcoll2d 44686* | cpcolld 44685 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
| ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑦 𝑥𝐹𝑦) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | ||
| Theorem | grucollcld 44687 | A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝐺) ⇒ ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺) | ||
| Theorem | ismnu 44688* |
The hypothesis of this theorem defines a class M of sets that we
temporarily call "minimal universes", and which will turn out
in
grumnueq 44714 to be exactly Grothendicek universes.
Minimal universes are
sets which satisfy the predicate on 𝑦 in rr-groth 44726, except for the
𝑥
∈ 𝑦 clause.
A minimal universe is closed under subsets (mnussd 44690), powersets (mnupwd 44694), and an operation which is similar to a combination of collection and union (mnuop3d 44698), from which closure under pairing (mnuprd 44703), unions (mnuunid 44704), and function ranges (mnurnd 44710) can be deduced, from which equivalence with Grothendieck universes (grumnueq 44714) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ 𝑀 ↔ ∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))) | ||
| Theorem | mnuop123d 44689* | Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝒫 𝐴 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))) | ||
| Theorem | mnussd 44690* | Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
| Theorem | mnuss2d 44691* | mnussd 44690 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑈) | ||
| Theorem | mnu0eld 44692* | A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑈) | ||
| Theorem | mnuop23d 44693* | Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) | ||
| Theorem | mnupwd 44694* | Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) | ||
| Theorem | mnusnd 44695* | Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → {𝐴} ∈ 𝑈) | ||
| Theorem | mnuprssd 44696* | A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) | ||
| Theorem | mnuprss2d 44697* | Special case of mnuprssd 44696. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ 𝐴 ⊆ 𝐶 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) | ||
| Theorem | mnuop3d 44698* | Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝑈 ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐹 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) | ||
| Theorem | mnuprdlem1 44699* | Lemma for mnuprd 44703. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}} & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑤) | ||
| Theorem | mnuprdlem2 44700* | Lemma for mnuprd 44703. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}} & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → ¬ 𝐴 = ∅) & ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑤) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |