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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremint-add02d 40701 Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (0 + 𝐴) = 𝐵)
 
Theoremint-sqgeq0d 40702 SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → 0 ≤ (𝐴 · 𝐵))
 
Theoremint-eqprincd 40703 PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))
 
Theoremint-eqtransd 40704 EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremint-eqmvtd 40705 EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = (𝐶 + 𝐷))       (𝜑𝐶 = (𝐵𝐷))
 
Theoremint-eqineqd 40706 EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐵𝐴)
 
Theoremint-ineqmvtd 40707 IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴 = (𝐶 + 𝐷))       (𝜑 → (𝐵𝐷) ≤ 𝐶)
 
Theoremint-ineq1stprincd 40708 FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐷𝐶)       (𝜑 → (𝐵 + 𝐷) ≤ (𝐴 + 𝐶))
 
Theoremint-ineq2ndprincd 40709 SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑 → 0 ≤ 𝐶)       (𝜑 → (𝐵 · 𝐶) ≤ (𝐴 · 𝐶))
 
Theoremint-ineqtransd 40710 InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐵)       (𝜑𝐶𝐴)
 
20.32.3  N-Digit Addition Proof Generator

This section formalizes theorems used in an n-digit addition proof generator.

Other theorems required: deccl 12091 addcomli 10809 00id 10792 addid1i 10804 addid2i 10805 eqid 2821 dec0h 12098 decadd 12130 decaddc 12131.

 
Theoremunitadd 40711 Theorem used in conjunction with decaddc 12131 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
 
20.32.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 40712 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵𝑈𝐵))) → (𝐺 Σg ⟨“𝑆𝑇𝑈”⟩) = (𝑆 + (𝑇 + 𝑈)))
 
Theoremgsumws4 40713 Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵 ∧ (𝑈𝐵𝑉𝐵)))) → (𝐺 Σg ⟨“𝑆𝑇𝑈𝑉”⟩) = (𝑆 + (𝑇 + (𝑈 + 𝑉))))
 
Theoremamgm2d 40714 Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 25554. (Contributed by Stanislas Polu, 8-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
 
Theoremamgm3d 40715 Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3))
 
Theoremamgm4d 40716 Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4))
 
20.33  Mathbox for Rohan Ridenour
 
20.33.1  Misc
 
TheoremspALT 40717 sp 2183 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2183 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremelnelneqd 40718 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.)
(𝜑𝐶𝐴)    &   (𝜑 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremelnelneq2d 40719 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.)
(𝜑𝐴𝐶)    &   (𝜑 → ¬ 𝐵𝐶)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremrr-spce 40720* Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝑉)       (𝜑 → ∃𝑥𝜓)
 
Theoremrexlimdvaacbv 40721* Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3271. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝑥 = 𝑦 → (𝜓𝜃))    &   ((𝜑 ∧ (𝑦𝐴𝜃)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimddvcbvw 40722* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40721. The equivalent of this theorem without the bound variable change is rexlimddv 3277. Version of rexlimddvcbv 40723 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
(𝜑 → ∃𝑥𝐴 𝜃)    &   ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜃𝜒))       (𝜑𝜓)
 
Theoremrexlimddvcbv 40723* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40721. The equivalent of this theorem without the bound variable change is rexlimddv 3277. Usage of this theorem is discouraged because it depends on ax-13 2391, see rexlimddvcbvw 40722 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜃)    &   ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜃𝜒))       (𝜑𝜓)
 
Theoremrr-elrnmpt3d 40724* Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝑉)    &   ((𝜑𝑥 = 𝐶) → 𝐵 = 𝐷)       (𝜑𝐷 ∈ ran 𝐹)
 
Theoremfndmexd 40725 If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
(𝜑𝐹𝑉)    &   (𝜑𝐹 Fn 𝐷)       (𝜑𝐷 ∈ V)
 
Theoremfinnzfsuppd 40726* If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.)
(𝜑𝐹𝑉)    &   (𝜑𝐹 Fn 𝐷)    &   (𝜑𝑍𝑈)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐷) → (𝑥𝐴 ∨ (𝐹𝑥) = 𝑍))       (𝜑𝐹 finSupp 𝑍)
 
Theoremrr-phpd 40727 Equivalent of php 8677 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ ω)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremsuceqd 40728 Deduction associated with suceq 6229. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → suc 𝐴 = suc 𝐵)
 
Theoremtfindsd 40729* Deduction associated with tfinds 7549. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = suc 𝑦 → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   ((𝜑𝑦 ∈ On ∧ 𝜃) → 𝜏)    &   ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)    &   (𝜑𝐴 ∈ On)       (𝜑𝜂)
 
20.33.2  Monoid rings
 
Syntaxcmnring 40730 Extend class notation with the monoid ring function.
class MndRing
 
Definitiondf-mnring 40731* 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(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))))⟩))
 
Theoremmnringvald 40732* Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   𝐴 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   𝐵 = (Base‘𝑉)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
 
Theoremmnringnmulrd 40733 Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ (.r‘ndx)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → (𝐸𝑉) = (𝐸𝐹))
 
Theoremmnringbased 40734 The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   𝐵 = (Base‘𝑉)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑𝐵 = (Base‘𝐹))
 
Theoremmnringbaserd 40735 The base set of a monoid ring. Converse of mnringbased 40734. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝐹)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑𝐵 = (Base‘𝑉))
 
Theoremmnringelbased 40736 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 = (0g𝑅)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → (𝑋𝐵 ↔ (𝑋 ∈ (𝐶m 𝐴) ∧ 𝑋 finSupp 0 )))
 
Theoremmnringbasefd 40737 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)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)    &   (𝜑𝑋𝐵)       (𝜑𝑋:𝐴𝐶)
 
Theoremmnringbasefsuppd 40738 Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝐹)    &    0 = (0g𝑅)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)    &   (𝜑𝑋𝐵)       (𝜑𝑋 finSupp 0 )
 
Theoremmnringaddgd 40739 The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → (+g𝑉) = (+g𝐹))
 
Theoremmnring0gd 40740 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)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → (0g𝑉) = (0g𝐹))
 
Theoremmnring0g2d 40741 The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &    0 = (0g𝑅)    &   𝐴 = (Base‘𝑀)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝑊)       (𝜑 → (𝐴 × { 0 }) = (0g𝐹))
 
Theoremmnringmulrd 40742* The ring product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝐹)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   𝐴 = (Base‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 ))))) = (.r𝐹))
 
Theoremmnringscad 40743 The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑𝑅 = (Scalar‘𝐹))
 
Theoremmnringvscad 40744 The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐵)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑 → ( ·𝑠𝑉) = ( ·𝑠𝐹))
 
Theoremmnringlmodd 40745 Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝑈)       (𝜑𝐹 ∈ LMod)
 
Theoremmnringmulrvald 40746* Value of multiplication in a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝐹)    &    = (.r𝑅)    &    𝟎 = (0g𝑅)    &   𝐴 = (Base‘𝑀)    &    + = (+g𝑀)    &    · = (.r𝐹)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋𝑎) (𝑌𝑏)), 𝟎 )))))
 
Theoremmnringmulrcld 40747 Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝐹)    &   𝐴 = (Base‘𝑀)    &    · = (.r𝐹)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 
20.33.3  Shorter primitive equivalent of ax-groth
 
20.33.3.1  Grothendieck universes are closed under collection
 
Theoremgru0eld 40748 A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → ∅ ∈ 𝐺)
 
Theoremgrusucd 40749 Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → suc 𝐴𝐺)
 
Theoremr1rankcld 40750 Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐴 ∈ (𝑅1𝑅))       (𝜑 → (rank‘𝐴) ∈ (𝑅1𝑅))
 
Theoremgrur1cld 40751 Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (𝑅1𝐴) ∈ 𝐺)
 
Theoremgrurankcld 40752 Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (rank‘𝐴) ∈ 𝐺)
 
Theoremgrurankrcld 40753 If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑 → (rank‘𝐴) ∈ 𝐺)    &   (𝜑𝐴𝑉)       (𝜑𝐴𝐺)
 
Syntaxcscott 40754 Extend class notation with the Scott's trick operation.
class Scott 𝐴
 
Definitiondf-scott 40755* 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 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
 
Theoremscotteqd 40756 Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → Scott 𝐴 = Scott 𝐵)
 
Theoremscotteq 40757 Closed form of scotteqd 40756. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵)
 
Theoremnfscott 40758 Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐴       𝑥Scott 𝐴
 
Theoremscottabf 40759* Value of the Scott operation at a class abstraction. Variant of scottab 40760 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottab 40760* Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottabes 40761* Value of the Scott operation at a class abstraction. Variant of scottab 40760 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottss 40762 Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Scott 𝐴𝐴
 
Theoremelscottab 40763* 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 → (𝜑𝜓))       (𝑦 ∈ Scott {𝑥𝜑} → 𝜓)
 
Theoremscottex2 40764 scottex 9290 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Scott 𝐴 ∈ V
 
Theoremscotteld 40765* The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑 → ∃𝑥 𝑥𝐴)       (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
 
Theoremscottelrankd 40766 Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐵 ∈ Scott 𝐴)    &   (𝜑𝐶 ∈ Scott 𝐴)       (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))
 
Theoremscottrankd 40767 Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐵 ∈ Scott 𝐴)       (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))
 
Theoremgruscottcld 40768 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.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐵𝐺)    &   (𝜑𝐵 ∈ Scott 𝐴)       (𝜑 → Scott 𝐴𝐺)
 
Syntaxccoll 40769 Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)
 
Definitiondf-coll 40770* 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) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
 
Theoremdfcoll2 40771* Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
 
Theoremcolleq12d 40772 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
 
Theoremcolleq1 40773 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))
 
Theoremcolleq2 40774 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵))
 
Theoremnfcoll 40775 Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹 Coll 𝐴)
 
Theoremcollexd 40776 The output of the collection operation is a set if the second input is. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐴𝑉)       (𝜑 → (𝐹 Coll 𝐴) ∈ V)
 
Theoremcpcolld 40777* Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
 
Theoremcpcoll2d 40778* cpcolld 40777 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑 → ∃𝑦 𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
 
Theoremgrucollcld 40779 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.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐹 ⊆ (𝐺 × 𝐺))    &   (𝜑𝐴𝐺)       (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺)
 
20.33.3.2  Minimal universes
 
Theoremismnu 40780* The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 40806 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 40818, except for the 𝑥𝑦 clause.

A minimal universe is closed under subsets (mnussd 40782), powersets (mnupwd 40786), and an operation which is similar to a combination of collection and union (mnuop3d 40790), from which closure under pairing (mnuprd 40795), unions (mnuunid 40796), and function ranges (mnurnd 40802) can be deduced, from which equivalence with Grothendieck universes (grumnueq 40806) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)

𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}       (𝑈𝑉 → (𝑈𝑀 ↔ ∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
 
Theoremmnuop123d 40781* Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
 
Theoremmnussd 40782* Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremmnuss2d 40783* mnussd 40782 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑 → ∃𝑥𝑈 𝐴𝑥)       (𝜑𝐴𝑈)
 
Theoremmnu0eld 40784* A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → ∅ ∈ 𝑈)
 
Theoremmnuop23d 40785* Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹𝑉)       (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
 
Theoremmnupwd 40786* Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)
 
Theoremmnusnd 40787* Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)
 
Theoremmnuprssd 40788* A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐶𝑈)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuprss2d 40789* Special case of mnuprssd 40788. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐶𝑈)    &   𝐴𝐶    &   𝐵𝐶       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuop3d 40790* Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹𝑈)       (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
 
Theoremmnuprdlem1 40791* Lemma for mnuprd 40795. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐴𝑤)
 
Theoremmnuprdlem2 40792* Lemma for mnuprd 40795. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐵𝑈)    &   (𝜑 → ¬ 𝐴 = ∅)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐵𝑤)
 
Theoremmnuprdlem3 40793* Lemma for mnuprd 40795. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   𝑖𝜑       (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
 
Theoremmnuprdlem4 40794* Lemma for mnuprd 40795. General case. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑 → ¬ 𝐴 = ∅)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuprd 40795* Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuunid 40796* Minimal universes are closed under union. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)
 
Theoremmnuund 40797* Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremmnutrcld 40798* Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremmnutrd 40799* Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)       (𝜑 → Tr 𝑈)
 
Theoremmnurndlem1 40800* Lemma for mnurnd 40802. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(𝜑𝐹:𝐴𝑈)    &   𝐴 ∈ V    &   (𝜑 → ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))       (𝜑 → ran 𝐹𝑤)
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