Theorem addrval 38670

Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
addrval	|- ( ( A e. C /\ B e. D ) -> ( A +r B ) = ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) )

Distinct variable groups:   v, A   v, B

Allowed substitution hints:   C( v)   D( v)

Proof of Theorem addrval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. C -> A e. _V )
2		elex 3212	. 2 |- ( B e. D -> B e. _V )
3		fveq1 6190	. . . . 5 |- ( x = A -> ( x ` v ) = ( A ` v ) )
4		fveq1 6190	. . . . 5 |- ( y = B -> ( y ` v ) = ( B ` v ) )
5	3, 4	oveqan12d 6669	. . . 4 |- ( ( x = A /\ y = B ) -> ( ( x ` v ) + ( y ` v ) ) = ( ( A ` v ) + ( B ` v ) ) )
6	5	mpteq2dv 4745	. . 3 |- ( ( x = A /\ y = B ) -> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) = ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) )
7		df-addr 38667	. . 3 |- +r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )
8		reex 10027	. . . 4 |- RR e. _V
9	8	mptex 6486	. . 3 |- ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) e. _V
10	6, 7, 9	ovmpt2a 6791	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A +r B ) = ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) )
11	1, 2, 10	syl2an 494	1 |- ( ( A e. C /\ B e. D ) -> ( A +r B ) = ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   + caddc 9939   +rcplusr 38661

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-addr 38667

This theorem is referenced by:  addrfv  38673  addrfn  38676