Theorem addrfn 38676

Description: Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
addrfn	|- ( ( A e. C /\ B e. D ) -> ( A +r B ) Fn RR )

Proof of Theorem addrfn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( ( A ` x ) + ( B ` x ) ) e. _V
2		eqid 2622	. . 3 |- ( x e. RR |-> ( ( A ` x ) + ( B ` x ) ) ) = ( x e. RR |-> ( ( A ` x ) + ( B ` x ) ) )
3	1, 2	fnmpti 6022	. 2 |- ( x e. RR |-> ( ( A ` x ) + ( B ` x ) ) ) Fn RR
4		addrval 38670	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A +r B ) = ( x e. RR |-> ( ( A ` x ) + ( B ` x ) ) ) )
5	4	fneq1d 5981	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( A +r B ) Fn RR <-> ( x e. RR |-> ( ( A ` x ) + ( B ` x ) ) ) Fn RR ) )
6	3, 5	mpbiri 248	1 |- ( ( A e. C /\ B e. D ) -> ( A +r B ) Fn RR )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   |-> cmpt 4729   Fn wfn 5883   ` 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:  addrcom  38679