Theorem dvhvaddval 36379

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

Hypothesis

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

Assertion

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

Distinct variable groups:   f, g, E   .+^ , f, g   T, f, g

Allowed substitution hints:   .+ ( f, g)   F( f, g)   G( f, g)

Proof of Theorem dvhvaddval

Dummy variables h i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( h = F -> ( 1st ` h ) = ( 1st ` F ) )
2	1	coeq1d 5283	. . 3 |- ( h = F -> ( ( 1st ` h ) o. ( 1st ` i ) ) = ( ( 1st ` F ) o. ( 1st ` i ) ) )
3		fveq2 6191	. . . 4 |- ( h = F -> ( 2nd ` h ) = ( 2nd ` F ) )
4	3	oveq1d 6665	. . 3 |- ( h = F -> ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) )
5	2, 4	opeq12d 4410	. 2 |- ( h = F -> <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. = <. ( ( 1st ` F ) o. ( 1st ` i ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) >. )
6		fveq2 6191	. . . 4 |- ( i = G -> ( 1st ` i ) = ( 1st ` G ) )
7	6	coeq2d 5284	. . 3 |- ( i = G -> ( ( 1st ` F ) o. ( 1st ` i ) ) = ( ( 1st ` F ) o. ( 1st ` G ) ) )
8		fveq2 6191	. . . 4 |- ( i = G -> ( 2nd ` i ) = ( 2nd ` G ) )
9	8	oveq2d 6666	. . 3 |- ( i = G -> ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) )
10	7, 9	opeq12d 4410	. 2 |- ( i = G -> <. ( ( 1st ` F ) o. ( 1st ` i ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` i ) ) >. = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )
11		dvhvaddval.a	. . 3 |- .+ = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
12	11	dvhvaddcbv 36378	. 2 |- .+ = ( h e. ( T X. E ) , i e. ( T X. E ) |-> <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. )
13		opex 4932	. 2 |- <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. e. _V
14	5, 10, 12, 13	ovmpt2 6796	1 |- ( ( F e. ( T X. E ) /\ G e. ( T X. E ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dvhvadd  36381  dvhopaddN  36403