Theorem plusffvalg 13659

Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
plusffvalg	|- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, .+ , y   x, V, y

Allowed substitution hints:   .+^ ( x, y)

Proof of Theorem plusffvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusffval.3	. 2 |- .+^ = ( +f ` G )
2		df-plusf 13652	. . 3 |- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
3		fveq2 5690	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		plusffval.1	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2289	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5690	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		plusffval.2	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2289	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 6092	. . . 4 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
10	5, 5, 9	mpoeq123dv 6140	. . 3 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) = ( x e. B , y e. B |-> ( x .+ y ) ) )
11		elex 2833	. . 3 |- ( G e. V -> G e. _V )
12		basfn 13389	. . . . . 6 |- Base Fn _V
13		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5478	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
15	12, 11, 14	sylancr 418	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
16	4, 15	eqeltrid 2325	. . . 4 |- ( G e. V -> B e. _V )
17		mpoexga 6438	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
18	16, 16, 17	syl2anc 415	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
19	2, 10, 11, 18	fvmptd3 5793	. 2 |- ( G e. V -> ( +f ` G ) = ( x e. B , y e. B |-> ( x .+ y ) ) )
20	1, 19	eqtrid 2283	1 |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   Basecbs 13330   +g cplusg 13408   +fcplusf 13650

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-plusf 13652

This theorem is referenced by:  plusfvalg  13660  plusfeqg  13661  plusffng  13662  mgmplusf  13663