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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremamgm4d 40901 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 40902 sp 2180 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2180 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremelnelneqd 40903 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 40904 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 40905* Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝑉)       (𝜑 → ∃𝑥𝜓)

Theoremrexlimdvaacbv 40906* Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3244. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝑥 = 𝑦 → (𝜓𝜃))    &   ((𝜑 ∧ (𝑦𝐴𝜃)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremrexlimddvcbvw 40907* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40906. The equivalent of this theorem without the bound variable change is rexlimddv 3250. Version of rexlimddvcbv 40908 with a disjoint variable condition, which does not require ax-13 2379. (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 40908* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40906. The equivalent of this theorem without the bound variable change is rexlimddv 3250. Usage of this theorem is discouraged because it depends on ax-13 2379, see rexlimddvcbvw 40907 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 40909* Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝑉)    &   ((𝜑𝑥 = 𝐶) → 𝐵 = 𝐷)       (𝜑𝐷 ∈ ran 𝐹)

Theoremfinnzfsuppd 40910* 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 40911 Equivalent of php 8685 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ ω)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)

Theoremsuceqd 40912 Deduction associated with suceq 6224. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → suc 𝐴 = suc 𝐵)

Theoremtfindsd 40913* Deduction associated with tfinds 7554. (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 40914 Extend class notation with the monoid ring function.
class MndRing

Definitiondf-mnring 40915* 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 40916* 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 40917 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 40918 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 40919 The base set of a monoid ring. Converse of mnringbased 40918. (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 40920 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 40921 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 40922 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 40923 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 40924 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 40925 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 40926* 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 40927 The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑𝑅𝑈)    &   (𝜑𝑀𝑊)       (𝜑𝑅 = (Scalar‘𝐹))

Theoremmnringvscad 40928 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 40929 Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝑈)       (𝜑𝐹 ∈ LMod)

Theoremmnringmulrvald 40930* 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 40931 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 40932 A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → ∅ ∈ 𝐺)

Theoremgrusucd 40933 Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → suc 𝐴𝐺)

Theoremr1rankcld 40934 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 40935 Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (𝑅1𝐴) ∈ 𝐺)

Theoremgrurankcld 40936 Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (rank‘𝐴) ∈ 𝐺)

Theoremgrurankrcld 40937 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 40938 Extend class notation with the Scott's trick operation.
class Scott 𝐴

Definitiondf-scott 40939* 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 40940 Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → Scott 𝐴 = Scott 𝐵)

Theoremscotteq 40941 Closed form of scotteqd 40940. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵)

Theoremnfscott 40942 Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐴       𝑥Scott 𝐴

Theoremscottabf 40943* Value of the Scott operation at a class abstraction. Variant of scottab 40944 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 40944* 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 40945* Value of the Scott operation at a class abstraction. Variant of scottab 40944 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Theoremscottss 40946 Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Scott 𝐴𝐴

Theoremelscottab 40947* 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 40948 scottex 9298 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Scott 𝐴 ∈ V

Theoremscotteld 40949* 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 40950 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 40951 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 40952 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 40953 Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)

Definitiondf-coll 40954* 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 40955* Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}

Theoremcolleq12d 40956 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 40957 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))

Theoremcolleq2 40958 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵))

Theoremnfcoll 40959 Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹 Coll 𝐴)

Theoremcollexd 40960 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 40961* Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)

Theoremcpcoll2d 40962* cpcolld 40961 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑 → ∃𝑦 𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)

Theoremgrucollcld 40963 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 40964* The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 40990 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 41002, except for the 𝑥𝑦 clause.

A minimal universe is closed under subsets (mnussd 40966), powersets (mnupwd 40970), and an operation which is similar to a combination of collection and union (mnuop3d 40974), from which closure under pairing (mnuprd 40979), unions (mnuunid 40980), and function ranges (mnurnd 40986) can be deduced, from which equivalence with Grothendieck universes (grumnueq 40990) 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 40965* 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 40966* 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 40967* mnussd 40966 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 40968* 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 40969* 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 40970* 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 40971* 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 40972* 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 40973* Special case of mnuprssd 40972. (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 40974* 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 40975* Lemma for mnuprd 40979. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐴𝑤)

Theoremmnuprdlem2 40976* Lemma for mnuprd 40979. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐵𝑈)    &   (𝜑 → ¬ 𝐴 = ∅)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐵𝑤)

Theoremmnuprdlem3 40977* Lemma for mnuprd 40979. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   𝑖𝜑       (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)

Theoremmnuprdlem4 40978* Lemma for mnuprd 40979. 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 40979* 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 40980* 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 40981* 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 40982* 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 40983* 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 40984* Lemma for mnurnd 40986. (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)

Theoremmnurndlem2 40985* Lemma for mnurnd 40986. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴𝑈)    &   𝐴 ∈ V       (𝜑 → ran 𝐹𝑈)

Theoremmnurnd 40986* Minimal universes contain ranges of functions from an element of the universe to the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴𝑈)       (𝜑 → ran 𝐹𝑈)

Theoremmnugrud 40987* Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)       (𝜑𝑈 ∈ Univ)

Theoremgrumnudlem 40988* Lemma for grumnud 40989. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝐺 ∈ Univ)    &   𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))    &   ((𝑖𝐺𝐺) → (𝑖𝐹 ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))    &   (( ∈ (𝐹 Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → ∃𝑢𝑓 (𝑖𝑢 𝑢 ∈ (𝐹 Coll 𝑧)))       (𝜑𝐺𝑀)

Theoremgrumnud 40989* Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝐺 ∈ Univ)       (𝜑𝐺𝑀)

Theoremgrumnueq 40990* The class of Grothendieck universes is equal to the class of minimal universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Univ = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}

20.33.3.3  Primitive equivalent of ax-groth

Theoremexpandan 40991 Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))

Theoremexpandexn 40992 Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑 ↔ ¬ 𝜓)       (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)

Theoremexpandral 40993 Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜓))

Theoremexpandrexn 40994 Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑 ↔ ¬ 𝜓)       (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴𝜓))

Theoremexpandrex 40995 Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴 → ¬ 𝜓))

Theoremexpanduniss 40996* Expand 𝐴𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
( 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))

Theoremismnuprim 40997* Express the predicate on 𝑈 in ismnu 40964 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))) ↔ ∀𝑧(𝑧𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑈 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑈 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤))))))))))))

Theoremrr-grothprimbi 40998* Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim 41003. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(∀𝑥𝑦 ∈ Univ 𝑥𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))

Theoreminagrud 40999 Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝐼 ∈ Inacc)       (𝜑 → (𝑅1𝐼) ∈ Univ)

Theoreminaex 41000* Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴𝑥)

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