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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsmlub 14901 Least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
SubGrp SubGrp SubGrp

Theoremlsmss1 14902 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss1b 14903 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss2 14904 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmss2b 14905 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp

Theoremlsmass 14906 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp

Theoremlsm01 14907 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremlsm02 14908 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremsubglsm 14909 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s                      SubGrp

Theoremlssnle 14910 Equivalent expressions for "not less than". (chnlei 21989 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp       SubGrp

Theoremlsmmod 14911 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 14912 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 14913* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremcntzrecd 14914 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp

Theoremlsmcntz 14915 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp       Cntz

Theoremlsmcntzr 14916 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp       Cntz

Theoremlsmdisj 14917 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2 14918 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3 14919 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp                            Cntz

Theoremlsmdisjr 14920 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2r 14921 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3r 14922 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
SubGrp       SubGrp       SubGrp                            Cntz

Theoremlsmdisj2a 14923 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj2b 14924 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp

Theoremlsmdisj3a 14925 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp              Cntz

Theoremlsmdisj3b 14926 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp       SubGrp       SubGrp              Cntz

Theoremsubgdisj1 14927 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 14928 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 14929 Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4182, 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 14930* 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 14931* 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 14932* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1f 14933 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj2f 14934 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1id 14935 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 14936 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 14937 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1rid 14938 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
Cntz       SubGrp       SubGrp

Theorempj1ghm 14939 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp                            s

Theorempj1ghm2 14940 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp                            s s

Theoremlsmhash 14941 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
Cntz       SubGrp       SubGrp

10.2.11  Free groups

Syntaxcefg 14942 Extend class notation with the free group equivalence relation.
~FG

Syntaxcfrgp 14943 Extend class notation with the free group construction.
freeGrp

Syntaxcvrgp 14944 Extend class notation with free group injection.
varFGrp

Definitiondf-efg 14945* 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 14946 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 14945. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp freeMnd s ~FG

Definitiondf-vrgp 14947* 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 14948* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremefgmf 14949* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremefgmnvl 14950* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)

Theoremefgrcl 14951 Lemma for efgval 14953. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        Word

Theoremefglem 14952* Lemma for efgval 14953. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        splice

Theoremefgval 14953* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG        splice

Theoremefger 14954 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG

Theoremefgi 14955 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgi0 14956 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgi1 14957 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Word        ~FG        splice

Theoremefgtf 14958* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice        splice

Theoremefgtval 14959* 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 14960* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice

Theoremefgi2 14961* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice

Theoremefgtlen 14962* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice

Theoremefginvrel2 14963* 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 14964* 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 14965* 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 14966* 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 14967* Value of the auxiliary function defining a sequence of extensions (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsdmi 14968* Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsval2 14969* 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 14970* 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 14971* A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgs1b 14972* Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgsp1 14973* 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 14974* An initial segment of an extension sequence is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^        ..^

Theoremefgsfo 14975* 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 14976* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlemf 14977* Lemma for efgredleme 14979. (Contributed by Mario Carneiro, 4-Jun-2016.)
Word        ~FG               splice               Word ..^

Theoremefgredlemg 14978* Lemma for efgred 14984. (Contributed by Mario Carneiro, 4-Jun-2016.)
Word        ~FG               splice               Word ..^

Theoremefgredleme 14979* Lemma for efgred 14984. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^                                                                                                                        substr concat substr

Theoremefgredlemd 14980* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^                                                                                                                        substr concat substr

Theoremefgredlemc 14981* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlemb 14982* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgredlem 14983* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgred 14984* The reduced word that forms the base of the sequence in efgsval 14967 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
Word        ~FG               splice               Word ..^

Theoremefgrelexlema 14985* 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 14986* 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 14987* 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 14988* There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgred2 14989* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        ~FG               splice               Word ..^

Theoremefgcpbllema 14990* Lemma for efgrelex 14987. 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 14991* Lemma for efgrelex 14987. 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 14992* 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

Theoremefgcpbl2 14993* 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

Theoremfrgpval 14994 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp       freeMnd        ~FG        s

Theoremfrgpcpbl 14995 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeGrp       freeMnd        ~FG

Theoremfrgp0 14996 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeGrp       ~FG

Theoremfrgpeccl 14997 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp       ~FG        Word

Theoremfrgpgrp 14998 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp

Theoremfrgpadd 14999 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        freeGrp       ~FG               concat

Theoremfrgpinv 15000* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Word        freeGrp       ~FG                      reverse

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