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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ex-bc 26701 | Example for df-bc 12952. (Contributed by AV, 4-Sep-2021.) |
⊢ (5C3) = ;10 | ||
Theorem | ex-hash 26702 | Example for df-hash 12980. (Contributed by AV, 4-Sep-2021.) |
⊢ (#‘{0, 1, 2}) = 3 | ||
Theorem | ex-sqrt 26703 | Example for df-sqrt 13823. (Contributed by AV, 4-Sep-2021.) |
⊢ (√‘;25) = 5 | ||
Theorem | ex-abs 26704 | Example for df-abs 13824. (Contributed by AV, 4-Sep-2021.) |
⊢ (abs‘-2) = 2 | ||
Theorem | ex-dvds 26705 | Example for df-dvds 14822: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ 3 ∥ 6 | ||
Theorem | ex-gcd 26706 | Example for df-gcd 15055. (Contributed by AV, 5-Sep-2021.) |
⊢ (-6 gcd 9) = 3 | ||
Theorem | ex-lcm 26707 | Example for df-lcm 15141. (Contributed by AV, 5-Sep-2021.) |
⊢ (6 lcm 9) = ;18 | ||
Theorem | ex-prmo 26708 | Example for df-prmo 15574: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (#p‘;10) = ;;210 | ||
Theorem | aevdemo 26709* | Proof illustrating the comment of aev2 1973. (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 26710 | 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 | avril1 26711 |
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 26712 | 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 26713 | The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. 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 put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ (ℎ ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑)) | ||
Theorem | 1p1e2apr1 26714 | 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 26715 |
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 26716 | 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 26717 | 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.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵) | ||
Syntax | cplig 26718 | Extend class notation with the class of all planar incidence geometries. |
class Plig | ||
Definition | df-plig 26719* |
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 us 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 𝑥 is defined as a set of lines 𝑙 satisfying three axioms. In the definition below, ∪ 𝑥 is the union of lines, that is, the plane, and 𝑎, 𝑏, 𝑐 denote points. Therefore, the axioms are: for all pairs of (distinct) points, there exists a unique line containing them; all lines contain at least two points; 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 26720* | The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.) |
⊢ 𝑃 = ∪ 𝐿 ⇒ ⊢ (𝐿 ∈ 𝐴 → (𝐿 ∈ Plig ↔ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐿 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐿 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐿 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)))) | ||
Theorem | tncp 26721* | In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.) |
⊢ 𝑃 = ∪ 𝐿 ⇒ ⊢ (𝐿 ∈ Plig → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐿 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)) | ||
Theorem | lpni 26722* | 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) | ||
Syntax | crpm 26723 | Ring primes. |
class RPrime | ||
Definition | df-rprm 26724* | Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 15263. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ RPrime = (𝑤 ∈ V ↦ ⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))}) | ||
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 26725 |
Alias for a1ii 1 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 1. (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ 𝜓 ⇒ ⊢ 𝜑 | ||
Theorem | id1 26726 |
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's not useful), and if so, delete it. Then repeat this recursively. One way to search for terminal theorems is to log the output ("open log x.txt") of "show usage WHATEVER" in metamath.exe 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 26727 | Extend class notation with the class of all group operations. |
class GrpOp | ||
Syntax | cgi 26728 | Extend class notation with a function mapping a group operation to the group's identity element. |
class GId | ||
Syntax | cgn 26729 | Extend class notation with a function mapping a group operation to the inverse function for the group. |
class inv | ||
Syntax | cgs 26730 | Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group. |
class /𝑔 | ||
Definition | df-grpo 26731* | 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 26732* | 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 26733* | 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 26734* | 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 26735* | 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 26736* | 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 26737 | 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 26738 | 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 26739* | 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 26740 | 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 26741 | 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 26742 | Lemma for grpoidinv 26746. (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 26743 | Lemma for grpoidinv 26746. (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 26744* | Lemma for grpoidinv 26746. (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 26745* | Lemma for grpoidinv 26746. (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 26746* | 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 26747* | 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 26748 | 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 26749 | The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ GrpOp | ||
Theorem | gidval 26750* | 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 26751* | Lemma for grpoidcl 26752 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 26752 | 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 26753* | 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 26754 | 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 26755 | 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 26756 | 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 26757* | 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 26758 | 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 26759 | 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 26760* | 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 26761* | 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 26762 | 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 26763 | 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 26764 | 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 26765 | 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 26766 | 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 26767 | 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 26768 | 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 26769 | 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 26770 | 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 26771 | 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 26772* | 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 26773 | 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 26774 | 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 26775 | 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 26776 | 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 26777 | 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 26778 | 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 26779 | 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 26780 | 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 26781 | Cancellation law for group division. (npcan 10169 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 26782 | Extend class notation with the class of all Abelian group operations. |
class AbelOp | ||
Definition | df-ablo 26783* | 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 26784* | 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 26785 | An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | ||
Theorem | ablocom 26786 | 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 26787 | 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 26788 | 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 26789* | 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 26790 | 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 26791 | 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 26792 | 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 26793 | 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 | ablonnncan 26794 | Cancellation law for group division. (nnncan 10195 analog.) (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))𝐷𝐶) = (𝐴𝐷𝐵)) | ||
Theorem | ablonncan 26795 | Cancellation law for group division. (nncan 10189 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 26796 | Cancellation law for group division. (nnncan1 10196 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 26797 | Extend class notation with the class of all complex vector spaces. |
class CVecOLD | ||
Definition | df-vc 26798* | 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 26799 | The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ Rel CVecOLD | ||
Theorem | vciOLD 26800* | Obsolete version of cvsi 22738 as of 21-Sep-2021. The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable 𝑊 was chosen because V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐺 = (1st ‘𝑊) & ⊢ 𝑆 = (2nd ‘𝑊) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))) |
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