Theorem dvhvaddval

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013)

	Ref	Expression
Hypothesis	dvhvaddval.a	|- .+ = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
Assertion	dvhvaddval	|- ( ( F e. ( T X. E ) /\ G e. ( T X. E ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )

Proof

Step	Hyp	Ref	Expression
1		dvhvaddval.a	|- .+ = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
2		fveq2	|- ( h = F -> ( 1st ` h ) = ( 1st ` F ) )
3	2	coeq1d	|- ( h = F -> ( ( 1st ` h ) o. ( 1st ` i ) ) = ( ( 1st ` F ) o. ( 1st ` i ) ) )
4		fveq2	|- ( h = F -> ( 2nd ` h ) = ( 2nd ` F ) )
5	4	oveq1d	|- ( h = F -> ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) )
6	3 5	opeq12d	|- ( h = F -> <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. = <. ( ( 1st ` F ) o. ( 1st ` i ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) >. )
7		fveq2	|- ( i = G -> ( 1st ` i ) = ( 1st ` G ) )
8	7	coeq2d	|- ( i = G -> ( ( 1st ` F ) o. ( 1st ` i ) ) = ( ( 1st ` F ) o. ( 1st ` G ) ) )
9		fveq2	|- ( i = G -> ( 2nd ` i ) = ( 2nd ` G ) )
10	9	oveq2d	|- ( i = G -> ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) )
11	8 10	opeq12d	|- ( i = G -> <. ( ( 1st ` F ) o. ( 1st ` i ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) >. = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )
12	1	dvhvaddcbv	|- .+ = ( h e. ( T X. E ) , i e. ( T X. E ) |-> <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. )
13		opex	|- <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. e. _V
14	6 11 12 13	ovmpo	|- ( ( F e. ( T X. E ) /\ G e. ( T X. E ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )