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Metamath Proof Explorer

Theorem qustrivr

Description: Converse of qustriv . (Contributed by Thierry Arnoux, 15-Jan-2024)

Ref Expression
Hypotheses qustrivr.1 
|- B = ( Base ` G )
qustrivr.2 
|- Q = ( G /s ( G ~QG H ) )
Assertion qustrivr 
|- ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> H = B )

Proof

Step Hyp Ref Expression
1 qustrivr.1 
 |-  B = ( Base ` G )
2 qustrivr.2 
 |-  Q = ( G /s ( G ~QG H ) )
3 2 a1i 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> Q = ( G /s ( G ~QG H ) ) )
4 1 a1i 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> B = ( Base ` G ) )
5 ovexd 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> ( G ~QG H ) e. _V )
6 simpl 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> G e. Grp )
7 3 4 5 6 qusbas 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> ( B /. ( G ~QG H ) ) = ( Base ` Q ) )
8 7 3adant3 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> ( B /. ( G ~QG H ) ) = ( Base ` Q ) )
9 simp3 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> ( Base ` Q ) = { H } )
10 8 9 eqtrd 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> ( B /. ( G ~QG H ) ) = { H } )
11 10 unieqd 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> U. ( B /. ( G ~QG H ) ) = U. { H } )
12 eqid 
 |-  ( G ~QG H ) = ( G ~QG H )
13 1 12 eqger 
 |-  ( H e. ( SubGrp ` G ) -> ( G ~QG H ) Er B )
14 13 adantl 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> ( G ~QG H ) Er B )
15 14 5 uniqs2 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) ) -> U. ( B /. ( G ~QG H ) ) = B )
16 15 3adant3 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> U. ( B /. ( G ~QG H ) ) = B )
17 unisng 
 |-  ( H e. ( SubGrp ` G ) -> U. { H } = H )
18 17 3ad2ant2 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> U. { H } = H )
19 11 16 18 3eqtr3rd 
 |-  ( ( G e. Grp /\ H e. ( SubGrp ` G ) /\ ( Base ` Q ) = { H } ) -> H = B )