mmtheorems104 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 10301-10400 - Page 104 of 107

Type	Label	Description
Statement
Theorem	cdj3lem2 10301	Lemma for cdj3 10307. Value of the first-component function S.
Theorem	cdj3lem2a 10302	Lemma for cdj3 10307. Closure of the first-component function S.
Theorem	cdj3lem2b 10303	Lemma for cdj3 10307. The first-component function S is bounded if the subspaces are completely disjoint.
Theorem	cdj3lem3 10304	Lemma for cdj3 10307. Value of the second-component function T.
Theorem	cdj3lem3a 10305	Lemma for cdj3 10307. Closure of the second-component function T.
Theorem	cdj3lem3b 10306	Lemma for cdj3 10307. The second-component function T is bounded if the subspaces are completely disjoint.
Theorem	cdj3 10307	Two ways to express "A and B are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520.
|- A e. SH   &   |- B e. SH   &   |- S = {<.x, y>. | (x e. (A +H B) /\ y = U.{z e. A | E.w e. B x = (z +h w)})}   &   |- T = {<.x, y>. | (x e. (A +H B) /\ y = U.{w e. B | E.z e. A x = (z +h w)})}   &   |- (ph <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))   &   |- (ps <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u))))   =>   |- (E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) <-> ((A i^i B) = 0H /\ ph /\ ps))
Sandboxes for user contributions
Sandbox guidelines
Theorem	sandbox 10308	(This theorem is a dummy placeholder for these guidelines.) "Sandboxes" are user-contributed sections that are not officially part of set.mm. They are included in the set.mm file in order to ensure that they are kept synchronized with label, definition, and theorem changes in set.mm. Eventually they may be broken out as separate modules, particularly in conjunction with any future Ghilbert translation. By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm. Sandboxes are provided as a courtesy to keep your work synchronized, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it. Notes: 1. I (N. Megill) have not necessarily checked definitions for soundness nor for agreement with the literature. In particular, a proof that 1 = 0 based on a sandbox definition will not considered in any challenge to prove set.mm inconsistent. (Such a proof will still be welcome, of course, so that the erroneous definition can be corrected.) 2. Over time I may decide to make a theorem "official," in which case it will be moved to the appropriate section of set.mm with an author acknowledgment. 3. At any time, I may revise definitions, theorems, proofs, and statement descriptions; add or delete theorems and/or definitions; or delete part or all of a sandbox if I feel it will not ultimately be useful or for any other reason. Guidelines: 1. If at all possible, please use only 0-ary class constants for new definitions, to make soundness checking easier. 2. Please try to follow the style of the rest of set.mm in terms of indentation, line length (79 characters or less), and comment markup (see HELP LANGUAGE in metamath.exe). Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description.
|- x = x
Sandbox for Paul Chapman
Miscellaneous theorems
Theorem	lemul2itALT 10309	Multiplication of both sides of 'less than or equal to' by a nonnegative number.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 <_ C)) -> (A <_ B -> (C x. A) <_ (C x. B)))
Theorem	lediv2itALT 10310	Division of both sides of 'less than or equal to' by a nonnegative number.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 <_ C)) -> (A <_ B -> (C / B) <_ (C / A)))
Theorem	abs2sqle 10311	The absolute values of two numbers compare as their squares.
|- A e. CC   &   |- B e. CC   =>   |- ((abs` A) <_ (abs` B) <-> ((abs` A)^2) <_ ((abs` B)^2))
Theorem	abs2sqlt 10312	The absolute values of two numbers compare as their squares.
|- A e. CC   &   |- B e. CC   =>   |- ((abs` A) < (abs` B) <-> ((abs` A)^2) < ((abs` B)^2))
Theorem	abs2sqlet 10313	The absolute values of two numbers compare as their squares.
|- ((A e. CC /\ B e. CC) -> ((abs` A) <_ (abs` B) <-> ((abs` A)^2) <_ ((abs` B)^2)))
Theorem	abs2sqltt 10314	The absolute values of two numbers compare as their squares.
|- ((A e. CC /\ B e. CC) -> ((abs` A) < (abs` B) <-> ((abs` A)^2) < ((abs` B)^2)))
Theorem	abs2dif 10315	Difference of absolute values.
|- A e. CC   &   |- B e. CC   =>   |- ((abs` A) - (abs` B)) <_ (abs` (A - B))
Theorem	abs2difabs 10316	Absolute value of difference of absolute values.
|- A e. CC   &   |- B e. CC   =>   |- (abs` ((abs` A) - (abs` B))) <_ (abs` (A - B))
Group homomorphism and isomorphism
Syntax	cghom 10317	Extend class notation to include the class of group homomorphisms.
class GrpHom
Syntax	cgiso 10318	Extend class notation to include the class of group isomorphisms.
class GrpIso
Definition	df-ghom 10319	Define the set of group homomorphisms from g to h.
|- GrpHom = {<.<.g, h>., s>. | ((g e. Grp /\ h e. Grp) /\ s = {f | (f:ran g-->ran h /\ A.x e. ran gA.y e. ran g((f` x)h(f` y)) = (f` (xgy)))})}
Definition	df-giso 10320	Define the set of group isomorphisms from g to h.
|- GrpIso = {<.<.g, h>., s>. | ((g e. Grp /\ h e. Grp) /\ s = {f e. (g GrpHom h) | f:ran g-1-1-onto->ran h})}
Theorem	elghomlem1 10321	Lemma for elghom 10323.
Theorem	elghomlem2 10322	Lemma for elghom 10323.
Theorem	elghom 10323	Membership in the set of group homomorphisms from G to H. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- W = ran H   =>   |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:X-->W /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))))
Theorem	ghomgrpilem1 10324	Lemma for ghomgrpi 10326.
Theorem	ghomgrpilem2 10325	Lemma for ghomgrpi 10326.
Theorem	ghomgrpi 10326	The image of a group homomorphism from G to H is a subgroup of H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
|- G e. Grp   &   |- H e. Grp   &   |- F e. (G GrpHom H)   &   |- Y = ran F   &   |- S = (H |` (Y X. Y))   =>   |- S e. (SubGrp` H)
Theorem	ghomsn 10327	The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
|- A e. V   &   |- G = {<.<.A, A>., A>.}   =>   |- (I |` {A}) e. (G GrpHom G)
Theorem	ghomgrplem 10328	Lemma for ghomgrp 10329.
Theorem	ghomgrp 10329	The image of a group homomorphism from G to H is a subgroup of H. (Contributed by Paul Chapman, 25-Feb-2008.)
|- Y = ran F   &   |- S = (H |` (Y X. Y))   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
Theorem	ghomfo 10330	A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
|- X = ran G   &   |- Y = ran F   &   |- S = (H |` (Y X. Y))   &   |- Z = ran S   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)
Theorem	ghomcl 10331	Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
|- X = ran G   &   |- Y = ran F   &   |- S = (H |` (Y X. Y))   &   |- Z = ran S   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (A e. X -> (F` A) e. Z))
Theorem	ghomlin 10332	Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
|- X = ran G   =>   |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (A e. X /\ B e. X)) -> ((F` A)H(F` B)) = (F` (AGB)))
Theorem	ghomid 10333	A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.)
|- U = (Id` G)   &   |- T = (Id` H)   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` U) = T)
Theorem	ghomgsg 10334	A group homomorphism from G to H is also a group homomorphism from G to its image in H. (Contributed by Paul Chapman, 3-Mar-2008.)
|- Y = ran F   &   |- S = (H |` (Y X. Y))   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))
Theorem	ghomf1olem 10335	Lemma for ghomf1o 10336.
Theorem	ghomf1o 10336	Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
|- X = ran G   &   |- Y = ran F   &   |- S = (H |` (Y X. Y))   &   |- Z = ran S   &   |- U = (Id` G)   &   |- T = (Id` H)   =>   |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-1-1-onto->Z <-> A.x e. X ((F` x) = T -> x = U)))
Theorem	elgiso 10337	Membership in the set of group isomorphisms from G to H. (Contributed by Paul Chapman, 25-Feb-2008.)
|- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpIso H) <-> (F e. (G GrpHom H) /\ F:ran G-1-1-onto->ran H)))
Symmetry groups and Cayley's Theorem
Syntax	csymgrp 10338	Extend class notation to include the class of symmetry groups.
class SymGrp
Definition	df-symgrp 10339	Define the symmetry group on set x. We represent the group as the set of 1-1-onto functions from x to itself under function composition.
|- SymGrp = {<.x, y>. | y = {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))}}
Theorem	elsymgrn 10340	Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.)
|- A e. V   &   |- P = {x | x:A-1-1-onto->A}   =>   |- (F e. P <-> F:A-1-1-onto->A)
Theorem	symgoprab 10341	Two ways to express the symmetry-group operator class abstraction. (Contributed by Paul Chapman, 25-Feb-2008.)
|- A e. V   &   |- P = {x | x:A-1-1-onto->A}   =>   |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}
Theorem	symgval 10342	The value of the symmetry group function at A. (Contributed by Paul Chapman, 25-Feb-2008.)
|- A e. V   &   |- P = {x | x:A-1-1-onto->A}   =>   |- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}
Theorem	symgoprval 10343	The value of the group operation of the symmetry group on A. (Contributed by Paul Chapman, 25-Feb-2008.)
|- A e. V   &   |- P = {x | x:A-1-1-onto->A}   =>   |- ((