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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsmcom2 14801 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
Cntz       SubGrp SubGrp

Theoremlsmub1 14802 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmub2 14803 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmunss 14804 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
SubGrp SubGrp

Theoremlsmless1 14805 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmless2 14806 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmless12 14807 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmidm 14808 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
SubGrp

Theoremlsmlub 14809 Least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
SubGrp SubGrp SubGrp

Theoremlsmss1 14810 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss1b 14811 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss2 14812 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss2b 14813 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmass 14814 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp

Theoremlsm01 14815 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremlsm02 14816 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremsubglsm 14817 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s                      SubGrp

Theoremlssnle 14818 Equivalent expressions for "not less than". (chnlei 21894 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp       SubGrp

Theoremlsmmod 14819 The modular law holds for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp

Theoremlsmmod2 14820 Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
SubGrp SubGrp SubGrp

Theoremlsmpropd 14821* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremcntzrecd 14822 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp

Theoremlsmcntz 14823 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp       Cntz

Theoremlsmcntzr 14824 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp       Cntz

Theoremlsmdisj 14825 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2 14826 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3 14827 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp                            Cntz

Theoremlsmdisjr 14828 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2r 14829 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3r 14830 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp                            Cntz

Theoremlsmdisj2a 14831 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2b 14832 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3a 14833 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp              Cntz

Theoremlsmdisj3b 14834 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp              Cntz

Theoremsubgdisj1 14835 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Cntz       SubGrp       SubGrp

Theoremsubgdisj2 14836 Vectors belonging to disjoint subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Cntz       SubGrp       SubGrp

Theoremsubgdisjb 14837 Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4138, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Cntz       SubGrp       SubGrp

Theorempj1fval 14838* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Theorempj1val 14839* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Theorempj1eu 14840* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1f 14841 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj2f 14842 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1id 14843 Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp

Theorempj1eq 14844 Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1lid 14845 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1rid 14846 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1ghm 14847 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp                            s

Theorempj1ghm2 14848 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp                            s s

Theoremlsmhash 14849 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp

10.2.11  Free groups

Syntaxcefg 14850 Extend class notation with the free group equivalence relation.
~FG

Syntaxcfrgp 14851 Extend class notation with the free group construction.
freeGrp

Syntaxcvrgp 14852 Extend class notation with free group injection.
varFGrp

Definitiondf-efg 14853* Define the free group equivalence relation, which is the smallest equivalence relation such that for any words and formal symbol with inverse , . (Contributed by Mario Carneiro, 1-Oct-2015.)
~FG Word Word splice

Definitiondf-frgp 14854 Define the free group on a set of generators, defined as the quotient of the free monoid on (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 14853. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp freeMnd s ~FG

Definitiondf-vrgp 14855* Define the canonical injection from the generating set into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
varFGrp ~FG

Theoremefgmval 14856* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremefgmf 14857* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremefgmnvl 14858* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)

Theoremefgrcl 14859 Lemma for efgval 14861. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        Word

Theoremefglem 14860* Lemma for efgval 14861. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        splice

Theoremefgval 14861* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG        splice

Theoremefger 14862 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG

Theoremefgi 14863 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgi0 14864 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgi1 14865 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgtf 14866* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice        splice

Theoremefgtval 14867* Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-2015.)
Word        ~FG               splice        splice

Theoremefgval2 14868* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice

Theoremefgi2 14869* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice

Theoremefgtlen 14870* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice

Theoremefginvrel2 14871* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice        concat reverse

Theoremefginvrel1 14872* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice        reverse concat

Theoremefgsf 14873* Value of the auxiliary function defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        Word ..^

Theoremefgsdm 14874* Elementhood in the domain of , the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^        Word ..^

Theoremefgsval 14875* Value of the auxiliary function defining a sequence of extensions (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsdmi 14876* Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsval2 14877* Value of the auxiliary function defining a sequence of extensions (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        Word concat concat

Theoremefgsrel 14878* The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgs1 14879* A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgs1b 14880* Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsp1 14881* If is an extension sequence and is an extension of the last element of , then is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        concat

Theoremefgsres 14882* An initial segment of an extension sequence is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        ..^

Theoremefgsfo 14883* For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indexes somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlema 14884* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlemf 14885* Lemma for efgredleme 14887. (Contributed by Mario Carneiro, 4-Jun-2016.)
Word        ~FG               splice               Word ..^

Theoremefgredlemg 14886* Lemma for efgred 14892. (Contributed by Mario Carneiro, 4-Jun-2016.)
Word        ~FG               splice               Word ..^

Theoremefgredleme 14887* Lemma for efgred 14892. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^                                                                                                                        substr concat substr

Theoremefgredlemd 14888* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^                                                                                                                        substr concat substr

Theoremefgredlemc 14889* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlemb 14890* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlem 14891* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgred 14892* The reduced word that forms the base of the sequence in efgsval 14875 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgrelexlema 14893* If two words are related under the free group equivalence, then there exist two extension sequences such that ends at , ends at , and and have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgrelexlemb 14894* If two words are related under the free group equivalence, then there exist two extension sequences such that ends at , ends at , and and have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgrelex 14895* If two words are related under the free group equivalence, then there exist two extension sequences such that ends at , ends at , and and have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredeu 14896* There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgred2 14897* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgcpbllema 14898* Lemma for efgrelex 14895. Define an auxiliary equivalence relation such that if there are sequences from to passing through the same reduced word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        concat concat concat concat        concat concat concat concat

Theoremefgcpbllemb 14899* Lemma for efgrelex 14895. Show that is an equivalence relation containing all direct extensions of a word, so is closed under . (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        concat concat concat concat

Theoremefgcpbl 14900* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        concat concat concat concat

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