P. 406 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40501-40600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gneispb 40501*	Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
Theorem	gneispace2 40502*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace3 40503*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace 40504*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))))
Theorem	gneispacef 40505*	A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacef2 40506*	A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
Theorem	gneispacefun 40507*	A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → Fun 𝐹)
Theorem	gneispacern 40508*	A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacern2 40509*	A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
Theorem	gneispace0nelrn 40510*	A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
Theorem	gneispace0nelrn2 40511*	A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅)
Theorem	gneispace0nelrn3 40512*	A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ ran 𝐹)
Theorem	gneispaceel 40513*	Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛)
Theorem	gneispaceel2 40514*	Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁)
Theorem	gneispacess 40515*	All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))
Theorem	gneispacess2 40516*	All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))
20.31.5  Exploring Higher Homotopy via Kerodon See https://kerodon.net/ for a work in progress by Jacob Lurie.
20.31.5.1  Simplicial Sets See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁.
Theorem	k0004lem1 40517	Application of ssin 4207 to range of a function. (Contributed by RP, 1-Apr-2021.)
⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷))
Theorem	k0004lem2 40518	A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑m 𝐴)))
Theorem	k0004lem3 40519	When the value of a mapping on a singleton is known, the mapping is a completely known singleton. (Contributed by RP, 2-Apr-2021.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∈ (𝐵 ↑m {𝐴}) ∧ (𝐹‘𝐴) = 𝐶) ↔ 𝐹 = {⟨𝐴, 𝐶⟩}))
Theorem	k0004val 40520*	The topological simplex of dimension 𝑁 is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.)
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})
Theorem	k0004ss1 40521*	The topological simplex of dimension 𝑁 is a subset of the real vectors of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1))))
Theorem	k0004ss2 40522*	The topological simplex of dimension 𝑁 is a subset of the base set of a real vector space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1)))))
Theorem	k0004ss3 40523*	The topological simplex of dimension 𝑁 is a subset of the base set of Euclidean space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(𝔼hil‘(𝑁 + 1))))
Theorem	k0004val0 40524*	The topological simplex of dimension 0 is a singleton. (Contributed by RP, 2-Apr-2021.)
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝐴‘0) = {{⟨1, 1⟩}}
20.32  Mathbox for Stanislas Polu
Theorem	inductionexd 40525	Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))
20.32.1  IMO Problems
20.32.1.1  IMO 1972 B2
Theorem	wwlemuld 40526	Natural deduction form of lemul2d 12476. (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 40527	Specialization of breq1d 5076 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 40528	Specialization of breq2d 5078 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 40529	Specialization of absmuld with absidd 14782. (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 40530	Natural dduction form of one side of imadisj 5948. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)
Theorem	imadisjlnd 40531	Natural deduction form of one negated side of imadisj 5948. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)
Theorem	wnefimgd 40532	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 40533	Natural deduction form of fco2 6533. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	wfximgfd 40534	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 40535*	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 40536*	Lemma for imo72b2 40545. (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 40537*	Natural deduction form of specialized suprleub 11607. (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 40538*	Lemma for imo72b2 40545. (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	syldbl2 40539	Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	suprlubrd 40540*	Natural deduction form of specialized suprlub 11605. (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 40541*	Lemma for imo72b2 40545. (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 40542	'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 40543	'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	rspcdvinvd 40544*	If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	imo72b2 40545*	IMO 1972 B2. (14th International Mathemahics 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)
20.32.2  INT Inequalities Proof Generator 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 10644 1red 10642 readdcld 10670 remulcld 10671 eqcomd 2827.
Theorem	int-addcomd 40546	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 40547	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 40548	AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 0 = (𝐴 − 𝐵))
Theorem	int-mulcomd 40549	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 40550	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 40551	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 40552	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 40553	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 40554	SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2))
Theorem	int-mul11d 40555	First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐵)
Theorem	int-mul12d 40556	Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐵)
Theorem	int-add01d 40557	First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 0) = 𝐵)
Theorem	int-add02d 40558	Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (0 + 𝐴) = 𝐵)
Theorem	int-sqgeq0d 40559	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 40560	PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))
Theorem	int-eqtransd 40561	EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	int-eqmvtd 40562	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 40563	EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝐴)
Theorem	int-ineqmvtd 40564	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 40565	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 40566	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 40567	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)
20.32.3  N-Digit Addition Proof Generator This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12114 addcomli 10832 00id 10815 addid1i 10827 addid2i 10828 eqid 2821 dec0h 12121 decadd 12153 decaddc 12154.
Theorem	unitadd 40568	Theorem used in conjunction with decaddc 12154 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)
Theorem	gsumws3 40569	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 40570	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 40571	Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 25567. (Contributed by Stanislas Polu, 8-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
Theorem	amgm3d 40572	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 40573	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))
20.33  Mathbox for Rohan Ridenour
20.33.1  Misc
Theorem	spALT 40574	sp 2182 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2182 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	elnelneqd 40575	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 40576	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 40577*	Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	rexlimdvaacbv 40578*	Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3285. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimddvcbvw 40579*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40578. The equivalent of this theorem without the bound variable change is rexlimddv 3291. Version of rexlimddvcbv 40580 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	rexlimddvcbv 40580*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40578. The equivalent of this theorem without the bound variable change is rexlimddv 3291. Usage of this theorem is discouraged because it depends on ax-13 2390, see rexlimddvcbvw 40579 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 40581*	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 40582	Equivalent of php 8701 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ω)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	suceqd 40583	Deduction associated with suceq 6256. (Contributed by Rohan Ridenour, 8-Aug-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → suc 𝐴 = suc 𝐵)
Theorem	tfindsd 40584*	Deduction associated with tfinds 7574. (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)
20.33.2  Shorter primitive equivalent of ax-groth
20.33.2.1  Grothendieck universes are closed under collection
Theorem	gru0eld 40585	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 40586	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 40587	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 40588	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 40589	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 40590	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 40591	Extend class notation with the Scott's trick operation.
class Scott 𝐴
Definition	df-scott 40592*	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 40593	Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)
Theorem	scotteq 40594	Closed form of scotteqd 40593. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵)
Theorem	nfscott 40595	Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Scott 𝐴
Theorem	scottabf 40596*	Value of the Scott operation at a class abstraction. Variant of scottab 40597 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 40597*	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 40598*	Value of the Scott operation at a class abstraction. Variant of scottab 40597 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 40599	Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Scott 𝐴 ⊆ 𝐴
Theorem	elscottab 40600*	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)