HomeTheorem List (p. 448 of 504)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | wnefimgd 44701 | 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 44702 | Natural deduction form of fco2 6714. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
| Theorem | wfximgfd 44703 | 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 44704* | 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 44705* | Lemma for imo72b2 44712. (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 44706* | Natural deduction form of specialized suprleub 12155. (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 44707* | Lemma for imo72b2 44712. (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 44708* | Natural deduction form of specialized suprlub 12153. (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 44709* | Lemma for imo72b2 44712. (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 44710 | '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 44711 | '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 44712* | 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 11181 1red 11179 readdcld 11208 remulcld 11209 eqcomd 2767. | ||
| Theorem | int-addcomd 44713 | 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 44714 | 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 44715 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 = (𝐴 − 𝐵)) | ||
| Theorem | int-mulcomd 44716 | 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 44717 | 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 44718 | 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 44719 | 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 44720 | 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 44721 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2)) | ||
| Theorem | int-mul11d 44722 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐵) | ||
| Theorem | int-mul12d 44723 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐵) | ||
| Theorem | int-add01d 44724 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐵) | ||
| Theorem | int-add02d 44725 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐵) | ||
| Theorem | int-sqgeq0d 44726 | 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 44727 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷)) | ||
| Theorem | int-eqtransd 44728 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
| Theorem | int-eqmvtd 44729 | 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 44730 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) | ||
| Theorem | int-ineqmvtd 44731 | 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 44732 | 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 44733 | 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 44734 | 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 12700 addcomli 11372 00id 11355 addridi 11367 addlidi 11368 eqid 2761 dec0h 12712 decadd 12744 decaddc 12745. | ||
| Theorem | unitadd 44735 | Theorem used in conjunction with decaddc 12745 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 44736 | 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 44737 | 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 44738 | Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 27031. (Contributed by Stanislas Polu, 8-Sep-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) | ||
| Theorem | amgm3d 44739 | 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 44740 | 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 44741 | sp 2217 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2217 instead. (New usage is discouraged.) |
| ⊢ (∀𝑥𝜑 → 𝜑) | ||
| Theorem | elnelneqd 44742 | 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 44743 | 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 44744* | Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
| ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
| Theorem | rexlimdvaacbv 44745* | Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3163. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
| Theorem | rexlimddvcbvw 44746* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44745. The equivalent of this theorem without the bound variable change is rexlimddv 3168. Version of rexlimddvcbv 44747 with a disjoint variable condition, which does not require ax-13 2402. (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 44747* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44745. The equivalent of this theorem without the bound variable change is rexlimddv 3168. Usage of this theorem is discouraged because it depends on ax-13 2402, see rexlimddvcbvw 44746 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 44748* | 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 44749 | Equivalent of php 9171 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | tfindsd 44750* | Deduction associated with tfinds 7836. (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 44751 | Extend class notation with the monoid ring function. |
| class MndRing | ||
| Definition | df-mnring 44752* | 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 44753* | 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 44754 | 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 44755 | 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 44756 | The base set of a monoid ring. Converse of mnringbased 44755. (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 44757 | 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 44758 | 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 44759 | 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 44760 | 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 44761 | 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 44762 | 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 44763* | 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 44764 | 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 44765 | 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 44766 | 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 44767* | 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 44768 | 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 44769 | 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 44770 | 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 44771 | 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 44772 | 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 44773 | 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 44774 | 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 44775 | Extend class notation with the Scott's trick operation. |
| class Scott 𝐴 | ||
| Definition | df-scott 44776* | 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 44777 | Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | scotteq 44778 | Closed form of scotteqd 44777. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵) | ||
| Theorem | nfscott 44779 | Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Scott 𝐴 | ||
| Theorem | scottabf 44780* | Value of the Scott operation at a class abstraction. Variant of scottab 44781 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 44781* | 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 44782* | Value of the Scott operation at a class abstraction. Variant of scottab 44781 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 44783 | Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Scott 𝐴 ⊆ 𝐴 | ||
| Theorem | elscottab 44784* | 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 44785 | scottex 9840 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
| ⊢ Scott 𝐴 ∈ V | ||
| Theorem | scotteld 44786* | 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 44787 | 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 44788 | 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 44789 | 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 44790 | Extend class notation with the collection operation. |
| class (𝐹 Coll 𝐴) | ||
| Definition | df-coll 44791* | 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 44792* | Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} | ||
| Theorem | colleq12d 44793 | 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 44794 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴)) | ||
| Theorem | colleq2 44795 | Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵)) | ||
| Theorem | nfcoll 44796 | Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹 Coll 𝐴) | ||
| Theorem | collexd 44797 | 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 44798* | 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 44799* | cpcolld 44798 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 44800 | 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) | ||
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