P. 452 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45101-45200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gneispacess 45101*	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 45102*	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))
21.38.8  Exploring Higher Homotopy via Kerodon See https://kerodon.net/ for a work in progress by Jacob Lurie.
21.38.8.1  Simplicial Sets See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁.
Theorem	k0004lem1 45103	Application of ssin 4184 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 45104	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 45105	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 45106*	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 45107*	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 45108*	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 45109*	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 45110*	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⟩}}
21.39  Mathbox for Stanislas Polu
Theorem	inductionexd 45111	Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))
21.39.1  IMO Problems
21.39.1.1  IMO 1972 B2
Theorem	wwlemuld 45112	Natural deduction form of lemul2d 13186. (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 45113	Specialization of breq1d 5113 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 45114	Specialization of breq2d 5115 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 45115	Specialization of absmuld with absidd 15567. (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 45116	Natural dduction form of one side of imadisj 6074. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)
Theorem	wnefimgd 45117	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 45118	Natural deduction form of fco2 6728. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	wfximgfd 45119	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 45120*	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 45121*	Lemma for imo72b2 45128. (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 45122*	Natural deduction form of specialized suprleub 12261. (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 45123*	Lemma for imo72b2 45128. (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 45124*	Natural deduction form of specialized suprlub 12259. (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 45125*	Lemma for imo72b2 45128. (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 45126	'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 45127	'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 45128*	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)
21.39.2  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 11289 1red 11287 readdcld 11316 remulcld 11317 eqcomd 2767.
Theorem	int-addcomd 45129	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 45130	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 45131	AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 0 = (𝐴 − 𝐵))
Theorem	int-mulcomd 45132	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 45133	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 45134	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 45135	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 45136	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 45137	SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2))
Theorem	int-mul11d 45138	First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐵)
Theorem	int-mul12d 45139	Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐵)
Theorem	int-add01d 45140	First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 0) = 𝐵)
Theorem	int-add02d 45141	Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (0 + 𝐴) = 𝐵)
Theorem	int-sqgeq0d 45142	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 45143	PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))
Theorem	int-eqtransd 45144	EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	int-eqmvtd 45145	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 45146	EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝐴)
Theorem	int-ineqmvtd 45147	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 45148	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 45149	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 45150	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)
21.39.3  N-Digit Addition Proof Generator This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12807 addcomli 11480 00id 11463 addridi 11475 addlidi 11476 eqid 2761 dec0h 12819 decadd 12851 decaddc 12852.
Theorem	unitadd 45151	Theorem used in conjunction with decaddc 12852 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
21.39.4  AM-GM (for k = 2,3,4)
Theorem	gsumws3 45152	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 45153	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 45154	Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 27296. (Contributed by Stanislas Polu, 8-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
Theorem	amgm3d 45155	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 45156	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))
21.40  Mathbox for Rohan Ridenour
21.40.1  Misc
Theorem	spALT 45157	sp 2220 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2220 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	rr-spce 45158*	Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	rexlimdvaacbv 45159*	Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3165. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimddvcbvw 45160*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 45159. The equivalent of this theorem without the bound variable change is rexlimddv 3170. Version of rexlimddvcbv 45161 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 45161*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 45159. The equivalent of this theorem without the bound variable change is rexlimddv 3170. Usage of this theorem is discouraged because it depends on ax-13 2402, see rexlimddvcbvw 45160 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 45162*	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 45163	Equivalent of php 9206 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ω)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	tfindsd 45164*	Deduction associated with tfinds 7860. (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)
21.40.2  Monoid rings
Syntax	cmnring 45165	Extend class notation with the monoid ring function.
class MndRing
Definition	df-mnring 45166*	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 45167*	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 45168	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 45169	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 45170	The base set of a monoid ring. Converse of mnringbased 45169. (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 45171	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 45172	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 45173	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 45174	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 45175	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 45176	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 45177*	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 45178	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 45179	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 45180	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 45181*	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 45182	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)
21.40.3  Shorter primitive equivalent of ax-groth
21.40.3.1  Grothendieck universes are closed under collection
Theorem	gru0eld 45183	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 45184	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 45185	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 45186	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 45187	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 45188	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)
Theorem	gruscottcld 45189	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 45190	Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)
Definition	df-coll 45191*	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 45192*	Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
Theorem	colleq12d 45193	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 45194	Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))
Theorem	colleq2 45195	Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵))
Theorem	nfcoll 45196	Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)
Theorem	collexd 45197	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 45198*	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 45199*	cpcolld 45198 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 45200	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)