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Metamath Proof Explorer

Theorem dpjeq

Description: Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016) (Revised by AV, 14-Jul-2019)

Ref Expression
Hypotheses dpjfval.1 
|- ( ph -> G dom DProd S )
dpjfval.2 
|- ( ph -> dom S = I )
dpjfval.p 
|- P = ( G dProj S )
dpjidcl.3 
|- ( ph -> A e. ( G DProd S ) )
dpjidcl.0 
|- .0. = ( 0g ` G )
dpjidcl.w 
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
dpjeq.c 
|- ( ph -> ( x e. I |-> C ) e. W )
Assertion dpjeq 
|- ( ph -> ( A = ( G gsum ( x e. I |-> C ) ) <-> A. x e. I ( ( P ` x ) ` A ) = C ) )

Proof

Step Hyp Ref Expression
1 dpjfval.1 
 |-  ( ph -> G dom DProd S )
2 dpjfval.2 
 |-  ( ph -> dom S = I )
3 dpjfval.p 
 |-  P = ( G dProj S )
4 dpjidcl.3 
 |-  ( ph -> A e. ( G DProd S ) )
5 dpjidcl.0 
 |-  .0. = ( 0g ` G )
6 dpjidcl.w 
 |-  W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
7 dpjeq.c 
 |-  ( ph -> ( x e. I |-> C ) e. W )
8 1 2 3 4 5 6 dpjidcl 
 |-  ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) )
9 8 simprd 
 |-  ( ph -> A = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) )
10 9 eqeq1d 
 |-  ( ph -> ( A = ( G gsum ( x e. I |-> C ) ) <-> ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) = ( G gsum ( x e. I |-> C ) ) ) )
11 8 simpld 
 |-  ( ph -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. W )
12 5 6 1 2 11 7 dprdf11 
 |-  ( ph -> ( ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) = ( G gsum ( x e. I |-> C ) ) <-> ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> C ) ) )
13 fvex 
 |-  ( ( P ` x ) ` A ) e. _V
14 13 rgenw 
 |-  A. x e. I ( ( P ` x ) ` A ) e. _V
15 mpteqb 
 |-  ( A. x e. I ( ( P ` x ) ` A ) e. _V -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> C ) <-> A. x e. I ( ( P ` x ) ` A ) = C ) )
16 14 15 mp1i 
 |-  ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> C ) <-> A. x e. I ( ( P ` x ) ` A ) = C ) )
17 10 12 16 3bitrd 
 |-  ( ph -> ( A = ( G gsum ( x e. I |-> C ) ) <-> A. x e. I ( ( P ` x ) ` A ) = C ) )