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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ex-prmo 29201 | Example for df-prmo 16838: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (#p‘;10) = ;;210 | ||
Theorem | aevdemo 29202* | Proof illustrating the comment of aev2 2061. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))) | ||
Theorem | ex-ind-dvds 29203 | Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.) |
⊢ (𝑁 ∈ ℕ0 → 3 ∥ ((4↑𝑁) + 2)) | ||
Theorem | ex-fpar 29204 | Formalized example provided in the comment for fpar 8036. (Contributed by AV, 3-Jan-2024.) |
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V))))) & ⊢ 𝐴 = (0[,)+∞) & ⊢ 𝐵 = ℝ & ⊢ 𝐹 = (√ ↾ 𝐴) & ⊢ 𝐺 = (sin ↾ 𝐵) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌))) | ||
Theorem | avril1 29205 |
Poisson d'Avril's Theorem. This theorem is noted for its
Selbstdokumentieren property, which means, literally,
"self-documenting" and recalls the principle of quidquid
german dictum
sit, altum viditur, often used in set theory. Starting with the
seemingly simple yet profound fact that any object 𝑥 equals
itself
(proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we
demonstrate that the power set of the real numbers, as a relation on the
value of the imaginary unit, does not conjoin with an empty relation on
the product of the additive and multiplicative identity elements,
leading to this startling conclusion that has left even seasoned
professional mathematicians scratching their heads. (Contributed by
Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.)
(New usage is discouraged.)
A reply to skeptics can be found at mmnotes.txt, under the 1-Apr-2006 entry. |
⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) | ||
Theorem | 2bornot2b 29206 | The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (2 · 𝐵 ∨ ¬ 2 · 𝐵) | ||
Theorem | helloworld 29207 | The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://helloworldcollection.de. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able to put it to rest with a remarkably short proof only four lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ (ℎ ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑)) | ||
Theorem | 1p1e2apr1 29208 | One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (1 + 1) = 2 | ||
Theorem | eqid1 29209 |
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐴 | ||
Theorem | 1div0apr 29210 | Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (1 / 0) = ∅ | ||
Theorem | topnfbey 29211 | Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Revised by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵) | ||
Theorem | 9p10ne21 29212 | 9 + 10 is not equal to 21. This disproves a popular meme which asserts that 9 + 10 does equal 21. See https://www.quora.com/Can-someone-try-to-prove-to-me-that-9+10-21 for attempts to prove that 9 + 10 = 21, and see https://tinyurl.com/9p10e21 for the history of the 9 + 10 = 21 meme. (Contributed by BTernaryTau, 25-Aug-2023.) |
⊢ (9 + ;10) ≠ ;21 | ||
Theorem | 9p10ne21fool 29213 | 9 + 10 equals 21. This astonishing thesis lives as a meme on the internet, and may be believed by quite some people. At least repeated requests to falsify it are a permanent part of the story. Prof. Loof Lirpa did not rest until he finally came up with a computer verifiable mathematical proof, that only a fool can think so. (Contributed by Prof. Loof Lirpa, 26-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((9 + ;10) = ;21 → 𝐹∅(0 · 1)) | ||
Syntax | cplig 29214 | Extend class notation with the class of all planar incidence geometries. |
class Plig | ||
Definition | df-plig 29215* |
Define the class of planar incidence geometries. We use Hilbert's
axioms and adapt them to planar geometry. We use ∈ for the
incidence relation. We could have used a generic binary relation, but
using ∈ allows to reuse previous results.
Much of what follows is
directly borrowed from Aitken, Incidence-Betweenness Geometry,
2008,
http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf.
The class Plig is the class of planar incidence geometries, where a planar incidence geometry is defined as a set of lines satisfying three axioms. In the definition below, 𝑥 denotes a planar incidence geometry, so ∪ 𝑥 denotes the union of its lines, that is, the set of points in the plane, 𝑙 denotes a line, and 𝑎, 𝑏, 𝑐 denote points. Therefore, the axioms are: 1) for all pairs of (distinct) points, there exists a unique line containing them; 2) all lines contain at least two points; 3) there exist three non-collinear points. (Contributed by FL, 2-Aug-2009.) |
⊢ Plig = {𝑥 ∣ (∀𝑎 ∈ ∪ 𝑥∀𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝑥 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝑥 ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥∃𝑐 ∈ ∪ 𝑥∀𝑙 ∈ 𝑥 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙))} | ||
Theorem | isplig 29216* | The predicate "is a planar incidence geometry" for sets. (Contributed by FL, 2-Aug-2009.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Plig ↔ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐺 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)))) | ||
Theorem | ispligb 29217* | The predicate "is a planar incidence geometry". (Contributed by BJ, 2-Dec-2021.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐺 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)))) | ||
Theorem | tncp 29218* | In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ (𝐺 ∈ Plig → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)) | ||
Theorem | l2p 29219* | For any line in a planar incidence geometry, there exist two different points on the line. (Contributed by AV, 28-Nov-2021.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ ((𝐺 ∈ Plig ∧ 𝐿 ∈ 𝐺) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿)) | ||
Theorem | lpni 29220* | For any line in a planar incidence geometry, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ ((𝐺 ∈ Plig ∧ 𝐿 ∈ 𝐺) → ∃𝑎 ∈ 𝑃 𝑎 ∉ 𝐿) | ||
Theorem | nsnlplig 29221 | There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.) |
⊢ (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺) | ||
Theorem | nsnlpligALT 29222 | Alternate version of nsnlplig 29221 using the predicate ∉ instead of ¬ ∈ and whose proof is shorter. (Contributed by AV, 5-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺) | ||
Theorem | n0lplig 29223 | There is no "empty line" in a planar incidence geometry. (Contributed by AV, 28-Nov-2021.) (Proof shortened by BJ, 2-Dec-2021.) |
⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | ||
Theorem | n0lpligALT 29224 | Alternate version of n0lplig 29223 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 29221. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺) | ||
Theorem | eulplig 29225* | Through two distinct points of a planar incidence geometry, there is a unique line. (Contributed by BJ, 2-Dec-2021.) |
⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ ((𝐺 ∈ Plig ∧ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ 𝐴 ≠ 𝐵)) → ∃!𝑙 ∈ 𝐺 (𝐴 ∈ 𝑙 ∧ 𝐵 ∈ 𝑙)) | ||
Theorem | pliguhgr 29226 | Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 27828 for a generalization of this case for arbitrary incidence structures (planar incidence geometries are such incidence structures). (Proposed by Gerard Lang, 24-Nov-2021.) (Contributed by AV, 28-Nov-2021.) |
⊢ (𝐺 ∈ Plig → ⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph) | ||
This section contains a few aliases that we temporarily keep to prevent broken links. If you land on any of these, please let the originating site and/or us know that the link that made you land here should be changed. | ||
Theorem | dummylink 29227 |
Alias for a1ii 2 that may be referenced in some older works, and
kept
here to prevent broken links.
If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to a1ii 2. (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 ⇒ ⊢ 𝜑 | ||
Theorem | id1 29228 |
Alias for idALT 23 that may be referenced in some older works, and
kept
here to prevent broken links.
If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to idALT 23. (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜑) | ||
The intent is for this deprecated section to be deleted once its theorems have extensible structure versions (or are not useful). You can make a list of "terminal" theorems (i.e., theorems not referenced by anything else) and for each theorem see if there exists an extensible structure version (or decide it is not useful), and if so, delete it. Then, repeat this recursively. One way to search for terminal theorems is to log the output ("MM> OPEN LOG xxx.txt") of "MM> SHOW USAGE <label-match>" in the Metamath program and search for "(None)". | ||
This section contains an earlier development of groups that was defined before extensible structures were introduced. The intent is for this deprecated section to be deleted once the corresponding definitions and theorems for complex topological vector spaces, which are using them, are revised accordingly. | ||
Syntax | cgr 29229 | Extend class notation with the class of all group operations. |
class GrpOp | ||
Syntax | cgi 29230 | Extend class notation with a function mapping a group operation to the group's identity element. |
class GId | ||
Syntax | cgn 29231 | Extend class notation with a function mapping a group operation to the inverse function for the group. |
class inv | ||
Syntax | cgs 29232 | Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group. |
class /𝑔 | ||
Definition | df-grpo 29233* | Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢))} | ||
Definition | df-gid 29234* | Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥))) | ||
Definition | df-ginv 29235* | Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.) |
⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔)))) | ||
Definition | df-gdiv 29236* | Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | ||
Theorem | isgrpo 29237* | The predicate "is a group operation." Note that 𝑋 is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))) | ||
Theorem | isgrpoi 29238* | Properties that determine a group operation. Read 𝑁 as 𝑁(𝑥). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 ∈ V & ⊢ 𝐺:(𝑋 × 𝑋)⟶𝑋 & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) & ⊢ 𝑈 ∈ 𝑋 & ⊢ (𝑥 ∈ 𝑋 → (𝑈𝐺𝑥) = 𝑥) & ⊢ (𝑥 ∈ 𝑋 → 𝑁 ∈ 𝑋) & ⊢ (𝑥 ∈ 𝑋 → (𝑁𝐺𝑥) = 𝑈) ⇒ ⊢ 𝐺 ∈ GrpOp | ||
Theorem | grpofo 29239 | A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) | ||
Theorem | grpocl 29240 | Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | ||
Theorem | grpolidinv 29241* | A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)) | ||
Theorem | grpon0 29242 | The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅) | ||
Theorem | grpoass 29243 | A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) | ||
Theorem | grpoidinvlem1 29244 | Lemma for grpoidinv 29248. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈) | ||
Theorem | grpoidinvlem2 29245 | Lemma for grpoidinv 29248. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌)) | ||
Theorem | grpoidinvlem3 29246* | Lemma for grpoidinv 29248. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝑋 (𝑈𝐺𝑥) = 𝑥) & ⊢ (𝜓 ↔ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ 𝑋 (𝑧𝐺𝑥) = 𝑈) ⇒ ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑈 ∈ 𝑋) ∧ (𝜑 ∧ 𝜓)) ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) | ||
Theorem | grpoidinvlem4 29247* | Lemma for grpoidinv 29248. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴)) | ||
Theorem | grpoidinv 29248* | A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))) | ||
Theorem | grpoideu 29249* | The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) | ||
Theorem | grporndm 29250 | A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺) | ||
Theorem | 0ngrp 29251 | The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ GrpOp | ||
Theorem | gidval 29252* | The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ 𝑉 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) | ||
Theorem | grpoidval 29253* | Lemma for grpoidcl 29254 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)) | ||
Theorem | grpoidcl 29254 | The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋) | ||
Theorem | grpoidinv2 29255* | A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))) | ||
Theorem | grpolid 29256 | The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴) | ||
Theorem | grporid 29257 | The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑈) = 𝐴) | ||
Theorem | grporcan 29258 | Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | grpoinveu 29259* | The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) | ||
Theorem | grpoid 29260 | Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴)) | ||
Theorem | grporn 29261 | The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) |
⊢ 𝐺 ∈ GrpOp & ⊢ dom 𝐺 = (𝑋 × 𝑋) ⇒ ⊢ 𝑋 = ran 𝐺 | ||
Theorem | grpoinvfval 29262* | The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))) | ||
Theorem | grpoinvval 29263* | The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)) | ||
Theorem | grpoinvcl 29264 | A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) | ||
Theorem | grpoinv 29265 | The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)) | ||
Theorem | grpolinv 29266 | The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈) | ||
Theorem | grporinv 29267 | The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈) | ||
Theorem | grpoinvid1 29268 | The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈)) | ||
Theorem | grpoinvid2 29269 | The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈)) | ||
Theorem | grpolcan 29270 | Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | grpo2inv 29271 | Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴) | ||
Theorem | grpoinvf 29272 | Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋) | ||
Theorem | grpoinvop 29273 | The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴))) | ||
Theorem | grpodivfval 29274* | Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) | ||
Theorem | grpodivval 29275 | Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵))) | ||
Theorem | grpodivinv 29276 | Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺𝐵)) | ||
Theorem | grpoinvdiv 29277 | Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝑁 = (inv‘𝐺) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴)) | ||
Theorem | grpodivf 29278 | Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋) | ||
Theorem | grpodivcl 29279 | Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋) | ||
Theorem | grpodivdiv 29280 | Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵))) | ||
Theorem | grpomuldivass 29281 | Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶))) | ||
Theorem | grpodivid 29282 | Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈) | ||
Theorem | grponpcan 29283 | Cancellation law for group division. (npcan 11343 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴) | ||
Syntax | cablo 29284 | Extend class notation with the class of all Abelian group operations. |
class AbelOp | ||
Definition | df-ablo 29285* | Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)} | ||
Theorem | isablo 29286* | The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) | ||
Theorem | ablogrpo 29287 | An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | ||
Theorem | ablocom 29288 | An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) | ||
Theorem | ablo32 29289 | Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵)) | ||
Theorem | ablo4 29290 | Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷))) | ||
Theorem | isabloi 29291* | Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) |
⊢ 𝐺 ∈ GrpOp & ⊢ dom 𝐺 = (𝑋 × 𝑋) & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) ⇒ ⊢ 𝐺 ∈ AbelOp | ||
Theorem | ablomuldiv 29292 | Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵)) | ||
Theorem | ablodivdiv 29293 | Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = ((𝐴𝐷𝐵)𝐺𝐶)) | ||
Theorem | ablodivdiv4 29294 | Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶))) | ||
Theorem | ablodiv32 29295 | Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵)) | ||
Theorem | ablonncan 29296 | Cancellation law for group division. (nncan 11363 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵) | ||
Theorem | ablonnncan1 29297 | Cancellation law for group division. (nnncan1 11370 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) |
⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵)) | ||
Syntax | cvc 29298 | Extend class notation with the class of all complex vector spaces. |
class CVecOLD | ||
Definition | df-vc 29299* | Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
⊢ CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} | ||
Theorem | vcrel 29300 | The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ Rel CVecOLD |
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