Metamath Proof Explorer

Theorem grplsm0l

Description: Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024)

Ref Expression
Hypotheses grplsm0l.b 
|- B = ( Base ` G )
grplsm0l.p 
|- .(+) = ( LSSum ` G )
grplsm0l.0 
|- .0. = ( 0g ` G )
Assertion grplsm0l 
|- ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( { .0. } .(+) A ) = A )

Proof

Step Hyp Ref Expression
1 grplsm0l.b 
 |-  B = ( Base ` G )
2 grplsm0l.p 
 |-  .(+) = ( LSSum ` G )
3 grplsm0l.0 
 |-  .0. = ( 0g ` G )
4 1 3 grpidcl 
 |-  ( G e. Grp -> .0. e. B )
5 4 snssd 
 |-  ( G e. Grp -> { .0. } C_ B )
6 eqid 
 |-  ( +g ` G ) = ( +g ` G )
7 1 6 2 lsmelvalx 
 |-  ( ( G e. Grp /\ { .0. } C_ B /\ A C_ B ) -> ( x e. ( { .0. } .(+) A ) <-> E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) ) )
8 7 3expa 
 |-  ( ( ( G e. Grp /\ { .0. } C_ B ) /\ A C_ B ) -> ( x e. ( { .0. } .(+) A ) <-> E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) ) )
9 8 an32s 
 |-  ( ( ( G e. Grp /\ A C_ B ) /\ { .0. } C_ B ) -> ( x e. ( { .0. } .(+) A ) <-> E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) ) )
10 5 9 mpidan 
 |-  ( ( G e. Grp /\ A C_ B ) -> ( x e. ( { .0. } .(+) A ) <-> E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) ) )
11 10 3adant3 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( x e. ( { .0. } .(+) A ) <-> E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) ) )
12 simpl1 
 |-  ( ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) /\ a e. A ) -> G e. Grp )
13 simp2 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> A C_ B )
14 13 sselda 
 |-  ( ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) /\ a e. A ) -> a e. B )
15 1 6 3 grplid 
 |-  ( ( G e. Grp /\ a e. B ) -> ( .0. ( +g ` G ) a ) = a )
16 12 14 15 syl2anc 
 |-  ( ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) /\ a e. A ) -> ( .0. ( +g ` G ) a ) = a )
17 16 eqeq2d 
 |-  ( ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) /\ a e. A ) -> ( x = ( .0. ( +g ` G ) a ) <-> x = a ) )
18 equcom 
 |-  ( x = a <-> a = x )
19 17 18 bitrdi 
 |-  ( ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) /\ a e. A ) -> ( x = ( .0. ( +g ` G ) a ) <-> a = x ) )
20 19 rexbidva 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( E. a e. A x = ( .0. ( +g ` G ) a ) <-> E. a e. A a = x ) )
21 3 fvexi 
 |-  .0. e. _V
22 oveq1 
 |-  ( o = .0. -> ( o ( +g ` G ) a ) = ( .0. ( +g ` G ) a ) )
23 22 eqeq2d 
 |-  ( o = .0. -> ( x = ( o ( +g ` G ) a ) <-> x = ( .0. ( +g ` G ) a ) ) )
24 23 rexbidv 
 |-  ( o = .0. -> ( E. a e. A x = ( o ( +g ` G ) a ) <-> E. a e. A x = ( .0. ( +g ` G ) a ) ) )
25 21 24 rexsn 
 |-  ( E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) <-> E. a e. A x = ( .0. ( +g ` G ) a ) )
26 risset 
 |-  ( x e. A <-> E. a e. A a = x )
27 20 25 26 3bitr4g 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( E. o e. { .0. } E. a e. A x = ( o ( +g ` G ) a ) <-> x e. A ) )
28 11 27 bitrd 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( x e. ( { .0. } .(+) A ) <-> x e. A ) )
29 28 eqrdv 
 |-  ( ( G e. Grp /\ A C_ B /\ A =/= (/) ) -> ( { .0. } .(+) A ) = A )