Metamath Proof Explorer

Theorem grpsubfval

Description: Group subtraction (division) operation. For a shorter proof using ax-rep , see grpsubfvalALT . (Contributed by NM, 31-Mar-2014) (Revised by Stefan O'Rear, 27-Mar-2015) Remove dependency on ax-rep . (Revised by Rohan Ridenour, 17-Aug-2023) (Proof shortened by AV, 19-Feb-2024)

Ref Expression
Hypotheses grpsubval.b 
|- B = ( Base ` G )
grpsubval.p 
|- .+ = ( +g ` G )
grpsubval.i 
|- I = ( invg ` G )
grpsubval.m 
|- .- = ( -g ` G )
Assertion grpsubfval 
|- .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )

Proof

Step Hyp Ref Expression
1 grpsubval.b 
 |-  B = ( Base ` G )
2 grpsubval.p 
 |-  .+ = ( +g ` G )
3 grpsubval.i 
 |-  I = ( invg ` G )
4 grpsubval.m 
 |-  .- = ( -g ` G )
5 fveq2 
 |-  ( g = G -> ( Base ` g ) = ( Base ` G ) )
6 5 1 eqtr4di 
 |-  ( g = G -> ( Base ` g ) = B )
7 fveq2 
 |-  ( g = G -> ( +g ` g ) = ( +g ` G ) )
8 7 2 eqtr4di 
 |-  ( g = G -> ( +g ` g ) = .+ )
9 eqidd 
 |-  ( g = G -> x = x )
10 fveq2 
 |-  ( g = G -> ( invg ` g ) = ( invg ` G ) )
11 10 3 eqtr4di 
 |-  ( g = G -> ( invg ` g ) = I )
12 11 fveq1d 
 |-  ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
13 8 9 12 oveq123d 
 |-  ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
14 6 6 13 mpoeq123dv 
 |-  ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
15 df-sbg 
 |-  -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
16 1 fvexi 
 |-  B e. _V
17 2 fvexi 
 |-  .+ e. _V
18 17 rnex 
 |-  ran .+ e. _V
19 p0ex 
 |-  { (/) } e. _V
20 18 19 unex 
 |-  ( ran .+ u. { (/) } ) e. _V
21 df-ov 
 |-  ( x .+ ( I ` y ) ) = ( .+ ` <. x , ( I ` y ) >. )
22 fvrn0 
 |-  ( .+ ` <. x , ( I ` y ) >. ) e. ( ran .+ u. { (/) } )
23 21 22 eqeltri 
 |-  ( x .+ ( I ` y ) ) e. ( ran .+ u. { (/) } )
24 23 rgen2w 
 |-  A. x e. B A. y e. B ( x .+ ( I ` y ) ) e. ( ran .+ u. { (/) } )
25 16 16 20 24 mpoexw 
 |-  ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V
26 14 15 25 fvmpt 
 |-  ( G e. _V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
27 4 26 syl5eq 
 |-  ( G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
28 fvprc 
 |-  ( -. G e. _V -> ( -g ` G ) = (/) )
29 4 28 syl5eq 
 |-  ( -. G e. _V -> .- = (/) )
30 fvprc 
 |-  ( -. G e. _V -> ( Base ` G ) = (/) )
31 1 30 syl5eq 
 |-  ( -. G e. _V -> B = (/) )
32 31 olcd 
 |-  ( -. G e. _V -> ( B = (/) \/ B = (/) ) )
33 0mpo0 
 |-  ( ( B = (/) \/ B = (/) ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = (/) )
34 32 33 syl 
 |-  ( -. G e. _V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = (/) )
35 29 34 eqtr4d 
 |-  ( -. G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
36 27 35 pm2.61i 
 |-  .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )