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Metamath Proof Explorer

Theorem qus0subgbas

Description: The base set of a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025)

Ref Expression
Hypotheses qus0subg.0 
|- .0. = ( 0g ` G )
qus0subg.s 
|- S = { .0. }
qus0subg.e 
|- .~ = ( G ~QG S )
qus0subg.u 
|- U = ( G /s .~ )
qus0subg.b 
|- B = ( Base ` G )
Assertion qus0subgbas 
|- ( G e. Grp -> ( Base ` U ) = { u | E. x e. B u = { x } } )

Proof

Step Hyp Ref Expression
1 qus0subg.0 
 |-  .0. = ( 0g ` G )
2 qus0subg.s 
 |-  S = { .0. }
3 qus0subg.e 
 |-  .~ = ( G ~QG S )
4 qus0subg.u 
 |-  U = ( G /s .~ )
5 qus0subg.b 
 |-  B = ( Base ` G )
6 df-qs 
 |-  ( B /. .~ ) = { u | E. x e. B u = [ x ] .~ }
7 4 a1i 
 |-  ( G e. Grp -> U = ( G /s .~ ) )
8 5 a1i 
 |-  ( G e. Grp -> B = ( Base ` G ) )
9 3 ovexi 
 |-  .~ e. _V
10 9 a1i 
 |-  ( G e. Grp -> .~ e. _V )
11 id 
 |-  ( G e. Grp -> G e. Grp )
12 7 8 10 11 qusbas 
 |-  ( G e. Grp -> ( B /. .~ ) = ( Base ` U ) )
13 1 2 5 3 eqg0subgecsn 
 |-  ( ( G e. Grp /\ x e. B ) -> [ x ] .~ = { x } )
14 13 eqeq2d 
 |-  ( ( G e. Grp /\ x e. B ) -> ( u = [ x ] .~ <-> u = { x } ) )
15 14 rexbidva 
 |-  ( G e. Grp -> ( E. x e. B u = [ x ] .~ <-> E. x e. B u = { x } ) )
16 15 abbidv 
 |-  ( G e. Grp -> { u | E. x e. B u = [ x ] .~ } = { u | E. x e. B u = { x } } )
17 6 12 16 3eqtr3a 
 |-  ( G e. Grp -> ( Base ` U ) = { u | E. x e. B u = { x } } )