P. 386 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)

Type	Label	Description
Statement
21.24.9  Heine-Borel Theorem
Theorem	heibor1lem 38501	Lemma for heibor1 38502. A compact metric space is complete. This proof works by considering the collection cls(𝐹 “ (ℤ≥‘𝑛)) for each 𝑛 ∈ ℕ, which has the finite intersection property because any finite intersection of upper integer sets is another upper integer set, so any finite intersection of the image closures will contain (𝐹 “ (ℤ≥‘𝑚)) for some 𝑚. Thus, by compactness, the intersection contains a point 𝑦, which must then be the convergent point of 𝐹. (Contributed by Jeff Madsen, 17-Jan-2014.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))    &   ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
Theorem	heibor1 38502	One half of heibor 38513, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 25515 and total boundedness here, which follows trivially from the fact that the set of all 𝑟-balls is an open cover of 𝑋, so finitely many cover 𝑋. (Contributed by Jeff Madsen, 16-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
Theorem	heiborlem1 38503*	Lemma for heibor 38513. We work with a fixed open cover 𝑈 throughout. The set 𝐾 is the set of all subsets of 𝑋 that admit no finite subcover of 𝑈. (We wish to prove that 𝐾 is empty.) If a set 𝐶 has no finite subcover, then any finite cover of 𝐶 must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)
Theorem	heiborlem2 38504*	Lemma for heibor 38513. Substitutions for the set 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Theorem	heiborlem3 38505*	Lemma for heibor 38513. Using countable choice ax-cc 10437, we have fixed in advance a collection of finite 2↑-𝑛 nets (𝐹‘𝑛) for 𝑋 (note that an 𝑟-net is a set of points in 𝑋 whose 𝑟 -balls cover 𝑋). The set 𝐺 is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set 𝐾). If the theorem was false, then 𝑋 would be in 𝐾, and so some ball at each level would also be in 𝐾. But we can say more than this; given a ball (𝑦𝐵𝑛) on level 𝑛, since level 𝑛 + 1 covers the space and thus also (𝑦𝐵𝑛), using heiborlem1 38503 there is a ball on the next level whose intersection with (𝑦𝐵𝑛) also has no finite subcover. Now since the set 𝐺 is a countable union of finite sets, it is countable (which needs ax-cc 10437 via iunctb 10577), and so we can apply ax-cc 10437 to 𝐺 directly to get a function from 𝐺 to itself, which points from each ball in 𝐾 to a ball on the next level in 𝐾, and such that the intersection between these balls is also in 𝐾. (Contributed by Jeff Madsen, 18-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    ⇒   ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
Theorem	heiborlem4 38506*	Lemma for heibor 38513. Using the function 𝑇 constructed in heiborlem3 38505, construct an infinite path in 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴)
Theorem	heiborlem5 38507*	Lemma for heibor 38513. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 25504. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)    ⇒   ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+))
Theorem	heiborlem6 38508*	Lemma for heibor 38513. Since the sequence of balls connected by the function 𝑇 ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most 3 / 2 times the size of the larger, and so if we expand each ball by a factor of 3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
Theorem	heiborlem7 38509*	Lemma for heibor 38513. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)    ⇒   ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟
Theorem	heiborlem8 38510*	Lemma for heibor 38513. The previous lemmas establish that the sequence 𝑀 is Cauchy, so using completeness we now consider the convergent point 𝑌. By assumption, 𝑈 is an open cover, so 𝑌 is an element of some 𝑍 ∈ 𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀‘𝑛) of the sequence is contained in 𝑍. And finally we arrive at a contradiction, because {𝑍} is a finite subcover of 𝑈 that covers ball(𝑀‘𝑛), yet ball(𝑀‘𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑 → 𝜓 where 𝜑 is all the accumulated hypotheses and 𝜓 is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ 𝑌 ∈ V    &   ⊢ (𝜑 → 𝑌 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	heiborlem9 38511*	Lemma for heibor 38513. Discharge the hypotheses of heiborlem8 38510 by applying caubl 25504 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))    &   ⊢ (𝜑 → 𝐶𝐺0)    &   ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → ∪ 𝑈 = 𝑋)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	heiborlem10 38512*	Lemma for heibor 38513. The last remaining piece of the proof is to find an element 𝐶 such that 𝐶𝐺0, i.e. 𝐶 is an element of (𝐹‘0) that has no finite subcover, which is true by heiborlem1 38503, since (𝐹‘0) is a finite cover of 𝑋, which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of 𝑈 that covers 𝑋, i.e. 𝑋 is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}    &   ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))    ⇒   ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣)
Theorem	heibor 38513	Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 38502 and heiborlem1 38503 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
21.24.10  Banach Fixed Point Theorem
Theorem	bfplem1 38514*	Lemma for bfp 38516. The sequence 𝐺, which simply starts from any point in the space and iterates 𝐹, satisfies the property that the distance from 𝐺(𝑛) to 𝐺(𝑛 + 1) decreases by at least 𝐾 after each step. Thus, the total distance from any 𝐺(𝑖) to 𝐺(𝑗) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point ((⇝𝑡‘𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐾 < 1)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴}))    ⇒   ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺))
Theorem	bfplem2 38515*	Lemma for bfp 38516. Using the point found in bfplem1 38514, we show that this convergent point is a fixed point of 𝐹. Since for any positive 𝑥, the sequence 𝐺 is in 𝐵(𝑥 / 2, 𝑃) for all 𝑘 ∈ (ℤ≥‘𝑗) (where 𝑃 = ((⇝𝑡‘𝐽)‘𝐺)), we have 𝐷(𝐺(𝑗 + 1), 𝐹(𝑃)) ≤ 𝐷(𝐺(𝑗), 𝑃) < 𝑥 / 2 and 𝐷(𝐺(𝑗 + 1), 𝑃) < 𝑥 / 2, so 𝐹(𝑃) is in every neighborhood of 𝑃 and 𝑃 is a fixed point of 𝐹. (Contributed by Jeff Madsen, 5-Jun-2014.)
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐾 < 1)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴}))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)
Theorem	bfp 38516*	Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if 𝐹 has two fixed points, then the distance between them is less than 𝐾 times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐾 < 1)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧)
21.24.11  Euclidean space
Syntax	crrn 38517	Extend class notation with the n-dimensional Euclidean space.
class ℝn
Definition	df-rrn 38518*	Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑m 𝑖), 𝑦 ∈ (ℝ ↑m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
Theorem	rrnval 38519*	The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
Theorem	rrnmval 38520*	The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝn‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	rrnmet 38521	Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (Met‘𝑋))
Theorem	rrndstprj1 38522	The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ (((𝐼 ∈ Fin ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹(ℝn‘𝐼)𝐺))
Theorem	rrndstprj2 38523*	Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 38522 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝn‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼))))
Theorem	rrncmslem 38524*	Lemma for rrncms 38525. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝐽 = (MetOpen‘(ℝn‘𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ∈ (Cau‘(ℝn‘𝐼)))    &   ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)    &   ⊢ 𝑃 = (𝑚 ∈ 𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝐹‘𝑡)‘𝑚))))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
Theorem	rrncms 38525	Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (CMet‘𝑋))
Theorem	repwsmet 38526	The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝑋 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋))
Theorem	rrnequiv 38527	The supremum metric on ℝ↑𝐼 is equivalent to the ℝn metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    ⇒   ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹𝐷𝐺) ≤ (𝐹(ℝn‘𝐼)𝐺) ∧ (𝐹(ℝn‘𝐼)𝐺) ≤ ((√‘(♯‘𝐼)) · (𝐹𝐷𝐺))))
Theorem	rrntotbnd 38528	A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝑀 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌))    ⇒   ⊢ (𝐼 ∈ Fin → (𝑀 ∈ (TotBnd‘𝑌) ↔ 𝑀 ∈ (Bnd‘𝑌)))
Theorem	rrnheibor 38529	Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑋 = (ℝ ↑m 𝐼)    &   ⊢ 𝑀 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌))    &   ⊢ 𝑇 = (MetOpen‘𝑀)    &   ⊢ 𝑈 = (MetOpen‘(ℝn‘𝐼))    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))
21.24.12  Intervals (continued)
Theorem	ismrer1 38530*	An isometry between ℝ and ℝ↑1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ (𝑅 Ismty (ℝn‘{𝐴})))
Theorem	reheibor 38531	Heine-Borel theorem for real numbers. A subset of ℝ is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑀 = ((abs ∘ − ) ↾ (𝑌 × 𝑌))    &   ⊢ 𝑇 = (MetOpen‘𝑀)    &   ⊢ 𝑈 = (topGen‘ran (,))    ⇒   ⊢ (𝑌 ⊆ ℝ → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))
Theorem	iccbnd 38532	A closed interval in ℝ is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐽 = (𝐴[,]𝐵)    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑀 ∈ (Bnd‘𝐽))
Theorem	icccmpALT 38533	A closed interval in ℝ is compact. Alternate proof of icccmp 25020 using the Heine-Borel theorem heibor 38513. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐽 = (𝐴[,]𝐵)    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))    &   ⊢ 𝑇 = (MetOpen‘𝑀)    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
21.24.13  Operation properties Obsolete part of JM's mathbox as of 13-Aug-2026. This and the following sections contain outdated material. All relevant definitions and theorems are available based on extensible structures, which is the preferred way in set.mm to deal with groups, rings and related structures. All definitions and theorems in the following are marked as deprecated (containing the tag "(New usage is discouraged.)" in their comments). In most cases, the corresponding definitions and theorems based on extensible structures are indicated within the comments. This and the following sections in JM's mathox will be removed from set.mm on 13-Aug-2027, unless there are any objections.
Syntax	cass 38534	Extend class notation with a device to add associativity to internal operations.
class Ass
Definition	df-ass 38535*	A device to add associativity to various sorts of internal operations. The definition is meaningful when 𝑔 is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
Syntax	cexid 38536	Extend class notation with the class of all the internal operations with an identity element.
class ExId
Definition	df-exid 38537*	A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
Theorem	isass 38538*	The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
Theorem	isexid 38539*	The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
21.24.14  Groups and related structures
Syntax	cmagm 38540	Extend class notation with the class of all magmas.
class Magma
Definition	df-mgmOLD 38541*	Obsolete version of df-mgm 18723 as of 3-Feb-2020. A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
Theorem	ismgmOLD 38542	Obsolete version of ismgm 18724 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
Theorem	clmgmOLD 38543	Obsolete version of mgmcl 18726 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Theorem	opidonOLD 38544	Obsolete version of mndpfo 18844 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
Theorem	rngopidOLD 38545	Obsolete version of mndpfo 18844 as of 23-Jan-2020. Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
Theorem	opidon2OLD 38546	Obsolete version of mndpfo 18844 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
Theorem	isexid2 38547*	Obsolete theorem. If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Theorem	exidu1 38548*	Obsolete theorem, use mgmidmo 18743 or mndideu 18832 instead. Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Theorem	idrval 38549*	Obsolete theorem, use grpidval 18744 instead. The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
Theorem	iorlid 38550	Obsolete theorem, use mgmidcl 18749 instead. A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    ⇒   ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋)
Theorem	cmpidelt 38551	Obsolete theorem, use mgmlrid 18750 instead. A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
Syntax	csem 38552	Extend class notation with the class of all semigroups.
class SemiGrp
Definition	df-sgrOLD 38553	Obsolete version of df-sgrp 18806 as of 3-Feb-2020. A semigroup is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
⊢ SemiGrp = (Magma ∩ Ass)
Theorem	smgrpismgmOLD 38554	Obsolete version of sgrpmgm 18811 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
Theorem	issmgrpOLD 38555*	Obsolete version of issgrp 18807 as of 3-Feb-2020. The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
Theorem	smgrpmgm 38556	Obsolete theorem, use sgrpmgm 18811 instead. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Theorem	smgrpassOLD 38557*	Obsolete version of sgrpass 18812 as of 3-Feb-2020. A semigroup is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ SemiGrp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
Syntax	cmndo 38558	Extend class notation with the class of all monoids.
class MndOp
Definition	df-mndo 38559	Obsolete definition, use df-mnd 18822 instead. A monoid is a semigroup with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
⊢ MndOp = (SemiGrp ∩ ExId )
Theorem	mndoissmgrpOLD 38560	Obsolete version of mndsgrp 18827 as of 3-Feb-2020. A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)
Theorem	mndoisexid 38561	Obsolete theorem, use mndid 18831 instead. A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
Theorem	mndoismgmOLD 38562	Obsolete version of mndmgm 18828 as of 3-Feb-2020. A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma)
Theorem	mndomgmid 38563	Obsolete theorem, use mndmgm 18828 and/or mndid 18831 instead. A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
Theorem	ismndo 38564*	Obsolete theorem, use ismnddef 18823 instead. The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
Theorem	ismndo1 38565*	Obsolete theorem, use ismnd 18824 instead. The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = dom dom 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
Theorem	ismndo2 38566*	Obsolete theorem, use ismnd 18824 instead. The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))
Theorem	grpomndo 38567	Obsolete theorem, use grpmnd 19038 instead. A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp)
Theorem	exidcl 38568	Obsolete theorem, use mgmcl 18726 instead. Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Theorem	exidreslem 38569*	Obsolete theorem, use 0gisid 18751 instead. Lemma for exidres 38570 and exidresid 38571. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
Theorem	exidres 38570	Obsolete theorem, use idressidex 18761 instead. The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId )
Theorem	exidresid 38571	Obsolete theorem, use idressid 18762 instead. The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))    ⇒   ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈)
Theorem	ablo4pnp 38572	Obsolete theorem, use ablsub4 19911 instead. A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
Theorem	grpoeqdivid 38573	Obsolete theorem, use grpsubeq0 19123 instead. Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))
Theorem	grposnOLD 38574	The group operation for the singleton group. Obsolete, use grp1 19144. instead. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
21.24.15  Group homomorphism and isomorphism
Syntax	cghomOLD 38575	Obsolete version of cghm 19314 as of 15-Mar-2020. Extend class notation to include the class of group homomorphisms. (New usage is discouraged.)
class GrpOpHom
Definition	df-ghomOLD 38576*	Obsolete version of df-ghm 19315 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to ℎ. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
Theorem	elghomlem1OLD 38577*	Obsolete as of 15-Mar-2020. Lemma for elghomOLD 38579. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Theorem	elghomlem2OLD 38578*	Obsolete as of 15-Mar-2020. Lemma for elghomOLD 38579. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Theorem	elghomOLD 38579*	Obsolete version of isghm 19317 as of 15-Mar-2020. Membership in the set of group homomorphisms from 𝐺 to 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑊 = ran 𝐻    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶𝑊 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Theorem	ghomlinOLD 38580	Obsolete version of ghmlin 19322 as of 15-Mar-2020. Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵)))
Theorem	ghomidOLD 38581	Obsolete version of ghmid 19323 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑈 = (GId‘𝐺)    &   ⊢ 𝑇 = (GId‘𝐻)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇)
Theorem	ghomf 38582	Obsolete theorem, use ghmf 19321 instead. Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑊 = ran 𝐻    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑊)
Theorem	ghomco 38583	Obsolete theorem, use ghmco 19337 instead. The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))
Theorem	ghomdiv 38584	Obsolete theorem, use ghmsub 19325 instead. Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    &   ⊢ 𝐶 = ( /𝑔 ‘𝐻)    ⇒   ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵)))
Theorem	grpokerinj 38585	Obsolete theorem, use kerf1ghm 19348 instead. A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑊 = (GId‘𝐺)    &   ⊢ 𝑌 = ran 𝐻    &   ⊢ 𝑈 = (GId‘𝐻)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊}))
21.24.16  Rings
Syntax	crngo 38586	Extend class notation with the class of all unital rings.
class RingOps
Definition	df-rngo 38587*	Obsolete defintion, use df-ring 20348 instead. Define the class of all unital rings. (Contributed by Jeff Hankins, 21-Nov-2006.) (New usage is discouraged.)
⊢ RingOps = {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))}
Theorem	relrngo 38588	Obsolete theorem. The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Rel RingOps
Theorem	isrngo 38589*	Obsolete theorem, use dfring2 20403 instead. The predicate "is a (unital) ring." Definition of "ring with unit" in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
Theorem	isrngod 38590*	Obsolete theorem, use isringd 20407 instead. Conditions that determine a ring. (Changed label from isringd 20407 to isrngod 38590-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐺 ∈ AbelOp)    &   ⊢ (𝜑 → 𝑋 = ran 𝐺)    &   ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)    ⇒   ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)
Theorem	rngoi 38591*	Obsolete theorem, use dfring2 20403 (left to right implication) instead. The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
Theorem	rngosm 38592	Obsolete theorem. Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
Theorem	rngocl 38593	Obsolete theorem, use ringcl 20363 instead. Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
Theorem	rngoid 38594*	Obsolete theorem, use ringid 20389 instead. The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
Theorem	rngoideu 38595*	Obsolete theorem, use ringideu 20367 instead. The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
Theorem	rngodi 38596	Obsolete theorem, use ringdi 20375 instead. Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
Theorem	rngodir 38597	Obsolete theorem, use ringdir 20376 instead. Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))
Theorem	rngoass 38598	Obsolete theorem, use ringass 20366 instead. Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
Theorem	rngo2 38599*	Obsolete theorem, use ringadd2 20391 instead. A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    &   ⊢ 𝐻 = (2nd ‘𝑅)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))
Theorem	rngoablo 38600	Obsolete theorem, use ringabl 20396 instead. A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑅)    ⇒   ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)