Metamath Proof Explorer

Theorem qusima

Description: The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024)

Ref Expression
Hypotheses qusima.b 
|- B = ( Base ` G )
qusima.q 
|- Q = ( G /s ( G ~QG N ) )
qusima.p 
|- .(+) = ( LSSum ` G )
qusima.e 
|- E = ( h e. S |-> ran ( x e. h |-> ( { x } .(+) N ) ) )
qusima.f 
|- F = ( x e. B |-> [ x ] ( G ~QG N ) )
qusima.n 
|- ( ph -> N e. ( NrmSGrp ` G ) )
qusima.h 
|- ( ph -> H e. S )
qusima.s 
|- ( ph -> S C_ ( SubGrp ` G ) )
Assertion qusima 
|- ( ph -> ( E ` H ) = ( F " H ) )

Proof

Step Hyp Ref Expression
1 qusima.b 
 |-  B = ( Base ` G )
2 qusima.q 
 |-  Q = ( G /s ( G ~QG N ) )
3 qusima.p 
 |-  .(+) = ( LSSum ` G )
4 qusima.e 
 |-  E = ( h e. S |-> ran ( x e. h |-> ( { x } .(+) N ) ) )
5 qusima.f 
 |-  F = ( x e. B |-> [ x ] ( G ~QG N ) )
6 qusima.n 
 |-  ( ph -> N e. ( NrmSGrp ` G ) )
7 qusima.h 
 |-  ( ph -> H e. S )
8 qusima.s 
 |-  ( ph -> S C_ ( SubGrp ` G ) )
9 5 reseq1i 
 |-  ( F |` H ) = ( ( x e. B |-> [ x ] ( G ~QG N ) ) |` H )
10 8 7 sseldd 
 |-  ( ph -> H e. ( SubGrp ` G ) )
11 1 subgss 
 |-  ( H e. ( SubGrp ` G ) -> H C_ B )
12 10 11 syl 
 |-  ( ph -> H C_ B )
13 12 resmptd 
 |-  ( ph -> ( ( x e. B |-> [ x ] ( G ~QG N ) ) |` H ) = ( x e. H |-> [ x ] ( G ~QG N ) ) )
14 nsgsubg 
 |-  ( N e. ( NrmSGrp ` G ) -> N e. ( SubGrp ` G ) )
15 6 14 syl 
 |-  ( ph -> N e. ( SubGrp ` G ) )
16 15 adantr 
 |-  ( ( ph /\ x e. H ) -> N e. ( SubGrp ` G ) )
17 12 sselda 
 |-  ( ( ph /\ x e. H ) -> x e. B )
18 1 3 16 17 quslsm 
 |-  ( ( ph /\ x e. H ) -> [ x ] ( G ~QG N ) = ( { x } .(+) N ) )
19 18 mpteq2dva 
 |-  ( ph -> ( x e. H |-> [ x ] ( G ~QG N ) ) = ( x e. H |-> ( { x } .(+) N ) ) )
20 13 19 eqtrd 
 |-  ( ph -> ( ( x e. B |-> [ x ] ( G ~QG N ) ) |` H ) = ( x e. H |-> ( { x } .(+) N ) ) )
21 9 20 syl5req 
 |-  ( ph -> ( x e. H |-> ( { x } .(+) N ) ) = ( F |` H ) )
22 21 adantr 
 |-  ( ( ph /\ h = H ) -> ( x e. H |-> ( { x } .(+) N ) ) = ( F |` H ) )
23 22 rneqd 
 |-  ( ( ph /\ h = H ) -> ran ( x e. H |-> ( { x } .(+) N ) ) = ran ( F |` H ) )
24 mpteq1 
 |-  ( h = H -> ( x e. h |-> ( { x } .(+) N ) ) = ( x e. H |-> ( { x } .(+) N ) ) )
25 24 rneqd 
 |-  ( h = H -> ran ( x e. h |-> ( { x } .(+) N ) ) = ran ( x e. H |-> ( { x } .(+) N ) ) )
26 25 adantl 
 |-  ( ( ph /\ h = H ) -> ran ( x e. h |-> ( { x } .(+) N ) ) = ran ( x e. H |-> ( { x } .(+) N ) ) )
27 df-ima 
 |-  ( F " H ) = ran ( F |` H )
28 27 a1i 
 |-  ( ( ph /\ h = H ) -> ( F " H ) = ran ( F |` H ) )
29 23 26 28 3eqtr4d 
 |-  ( ( ph /\ h = H ) -> ran ( x e. h |-> ( { x } .(+) N ) ) = ( F " H ) )
30 1 fvexi 
 |-  B e. _V
31 30 mptex 
 |-  ( x e. B |-> [ x ] ( G ~QG N ) ) e. _V
32 5 31 eqeltri 
 |-  F e. _V
33 imaexg 
 |-  ( F e. _V -> ( F " H ) e. _V )
34 32 33 mp1i 
 |-  ( ph -> ( F " H ) e. _V )
35 4 29 7 34 fvmptd2 
 |-  ( ph -> ( E ` H ) = ( F " H ) )