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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | wwlemuld 44601 | Natural deduction form of lemul2d 13028. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) & ⊢ (𝜑 → 0 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
| Theorem | leeq1d 44602 | Specialization of breq1d 5089 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐶) | ||
| Theorem | leeq2d 44603 | Specialization of breq2d 5091 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐷) | ||
| Theorem | absmulrposd 44604 | Specialization of absmuld with absidd 15383. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = (𝐴 · (abs‘𝐵))) | ||
| Theorem | imadisjld 44605 | Natural dduction form of one side of imadisj 6039. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅) | ||
| Theorem | wnefimgd 44606 | The image of a mapping from A is nonempty if A is nonempty. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) | ||
| Theorem | fco2d 44607 | Natural deduction form of fco2 6688. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
| Theorem | wfximgfd 44608 | The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴)) | ||
| Theorem | extoimad 44609* | If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶) | ||
| Theorem | imo72b2lem0 44610* | Lemma for imo72b2 44617. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) ⇒ ⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < )) | ||
| Theorem | suprleubrd 44611* | Natural deduction form of specialized suprleub 12120. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵) | ||
| Theorem | imo72b2lem2 44612* | Lemma for imo72b2 44617. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹‘𝑧)) ≤ 𝐶) ⇒ ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) | ||
| Theorem | suprlubrd 44613* | Natural deduction form of specialized suprlub 12118. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝐵 < 𝑧) ⇒ ⊢ (𝜑 → 𝐵 < sup(𝐴, ℝ, < )) | ||
| Theorem | imo72b2lem1 44614* | Lemma for imo72b2 44617. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) ⇒ ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) | ||
| Theorem | lemuldiv3d 44615 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) | ||
| Theorem | lemuldiv4d 44616 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) | ||
| Theorem | imo72b2 44617* | IMO 1972 B2. (14th International Mathematical Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) ⇒ ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1) | ||
This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf. Other theorems required: 0red 11145 1red 11143 readdcld 11172 remulcld 11173 eqcomd 2746. | ||
| Theorem | int-addcomd 44618 | AdditionCommutativity 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-addassocd 44619 | 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 44620 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 = (𝐴 − 𝐵)) | ||
| Theorem | int-mulcomd 44621 | 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 44622 | 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 44623 | 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 44624 | 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 44625 | 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 44626 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2)) | ||
| Theorem | int-mul11d 44627 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐵) | ||
| Theorem | int-mul12d 44628 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐵) | ||
| Theorem | int-add01d 44629 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐵) | ||
| Theorem | int-add02d 44630 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐵) | ||
| Theorem | int-sqgeq0d 44631 | 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 44632 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷)) | ||
| Theorem | int-eqtransd 44633 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
| Theorem | int-eqmvtd 44634 | 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 44635 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) | ||
| Theorem | int-ineqmvtd 44636 | 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 44637 | 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 44638 | 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 44639 | 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 12657 addcomli 11336 00id 11319 addridi 11331 addlidi 11332 eqid 2740 dec0h 12664 decadd 12696 decaddc 12697. | ||
| Theorem | unitadd 44640 | Theorem used in conjunction with decaddc 12697 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 44641 | 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 44642 | 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 44643 | Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 26978. (Contributed by Stanislas Polu, 8-Sep-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) | ||
| Theorem | amgm3d 44644 | 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 44645 | 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 44646 | sp 2195 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2195 instead. (New usage is discouraged.) |
| ⊢ (∀𝑥𝜑 → 𝜑) | ||
| Theorem | elnelneqd 44647 | 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 44648 | 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 44649* | Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
| ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
| Theorem | rexlimdvaacbv 44650* | Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3142. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
| Theorem | rexlimddvcbvw 44651* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44650. The equivalent of this theorem without the bound variable change is rexlimddv 3147. Version of rexlimddvcbv 44652 with a disjoint variable condition, which does not require ax-13 2380. (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 44652* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44650. The equivalent of this theorem without the bound variable change is rexlimddv 3147. Usage of this theorem is discouraged because it depends on ax-13 2380, see rexlimddvcbvw 44651 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 44653* | 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 44654 | Equivalent of php 9138 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | tfindsd 44655* | Deduction associated with tfinds 7807. (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 44656 | Extend class notation with the monoid ring function. |
| class MndRing | ||
| Definition | df-mnring 44657* | 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 44658* | 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 44659 | 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 44660 | 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 44661 | The base set of a monoid ring. Converse of mnringbased 44660. (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 44662 | 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 44663 | 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 44664 | 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 44665 | 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 44666 | 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 44667 | 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 44668* | 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 44669 | 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 44670 | 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 44671 | 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 44672* | 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 44673 | 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 44674 | 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 44675 | 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 44676 | 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 44677 | 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 44678 | 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 44679 | 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 44680 | Extend class notation with the Scott's trick operation. |
| class Scott 𝐴 | ||
| Definition | df-scott 44681* | 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 44682 | Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | scotteq 44683 | Closed form of scotteqd 44682. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | nfscott 44684 | Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Scott 𝐴 | ||
| Theorem | scottabf 44685* | Value of the Scott operation at a class abstraction. Variant of scottab 44686 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 44686* | 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 44687* | Value of the Scott operation at a class abstraction. Variant of scottab 44686 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 44688 | Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Scott 𝐴 ⊆ 𝐴 | ||
| Theorem | elscottab 44689* | 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 44690 | scottex 9807 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ Scott 𝐴 ∈ V | ||
| Theorem | scotteld 44691* | 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 44692 | 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 44693 | 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 44694 | 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 44695 | Extend class notation with the collection operation. |
| class (𝐹 Coll 𝐴) | ||
| Definition | df-coll 44696* | 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 44697* | Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} | ||
| Theorem | colleq12d 44698 | 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 44699 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴)) | ||
| Theorem | colleq2 44700 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵)) | ||
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