Theorem plusfeqg 13066

Description: If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
plusffval.1	|- B = ( Base ` G )
plusffval.2	|- .+ = ( +g ` G )
plusffval.3	|- .+^ = ( +f ` G )

Assertion

Ref	Expression
plusfeqg	|- ( ( G e. V /\ .+ Fn ( B X. B ) ) -> .+^ = .+ )

Proof of Theorem plusfeqg

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

Step	Hyp	Ref	Expression
1		plusffval.1	. . . 4 |- B = ( Base ` G )
2		plusffval.2	. . . 4 |- .+ = ( +g ` G )
3		plusffval.3	. . . 4 |- .+^ = ( +f ` G )
4	1, 2, 3	plusffvalg 13064	. . 3 |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )
5	4	adantr 276	. 2 |- ( ( G e. V /\ .+ Fn ( B X. B ) ) -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )
6		fnovim 6035	. . 3 |- ( .+ Fn ( B X. B ) -> .+ = ( x e. B , y e. B |-> ( x .+ y ) ) )
7	6	adantl 277	. 2 |- ( ( G e. V /\ .+ Fn ( B X. B ) ) -> .+ = ( x e. B , y e. B |-> ( x .+ y ) ) )
8	5, 7	eqtr4d 2232	1 |- ( ( G e. V /\ .+ Fn ( B X. B ) ) -> .+^ = .+ )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   X. cxp 4662   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   Basecbs 12703   +g cplusg 12780   +fcplusf 13055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-plusf 13057

This theorem is referenced by:  mgmb1mgm1  13070  mndfo  13141  cnfldplusf  14206