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Metamath Proof Explorer

Theorem vafval

Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007) (New usage is discouraged.)

Ref Expression
Hypothesis vafval.2 
|- G = ( +v ` U )
Assertion vafval 
|- G = ( 1st ` ( 1st ` U ) )

Proof

Step Hyp Ref Expression
1 vafval.2 
 |-  G = ( +v ` U )
2 df-va 
 |-  +v = ( 1st o. 1st )
3 2 fveq1i 
 |-  ( +v ` U ) = ( ( 1st o. 1st ) ` U )
4 fo1st 
 |-  1st : _V -onto-> _V
5 fof 
 |-  ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6 4 5 ax-mp 
 |-  1st : _V --> _V
7 fvco3 
 |-  ( ( 1st : _V --> _V /\ U e. _V ) -> ( ( 1st o. 1st ) ` U ) = ( 1st ` ( 1st ` U ) ) )
8 6 7 mpan 
 |-  ( U e. _V -> ( ( 1st o. 1st ) ` U ) = ( 1st ` ( 1st ` U ) ) )
9 3 8 eqtrid 
 |-  ( U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
10 fvprc 
 |-  ( -. U e. _V -> ( +v ` U ) = (/) )
11 fvprc 
 |-  ( -. U e. _V -> ( 1st ` U ) = (/) )
12 11 fveq2d 
 |-  ( -. U e. _V -> ( 1st ` ( 1st ` U ) ) = ( 1st ` (/) ) )
13 1st0 
 |-  ( 1st ` (/) ) = (/)
14 12 13 eqtr2di 
 |-  ( -. U e. _V -> (/) = ( 1st ` ( 1st ` U ) ) )
15 10 14 eqtrd 
 |-  ( -. U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
16 9 15 pm2.61i 
 |-  ( +v ` U ) = ( 1st ` ( 1st ` U ) )
17 1 16 eqtri 
 |-  G = ( 1st ` ( 1st ` U ) )