Theorem plusffvalg 12835

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 12828	. . 3 |- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
3		fveq2 5534	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		plusffval.1	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2240	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5534	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		plusffval.2	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2240	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 5912	. . . 4 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
10	5, 5, 9	mpoeq123dv 5957	. . 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 2763	. . 3 |- ( G e. V -> G e. _V )
12		basfn 12569	. . . . . 6 |- Base Fn _V
13		funfvex 5551	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5335	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
15	12, 11, 14	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
16	4, 15	eqeltrid 2276	. . . 4 |- ( G e. V -> B e. _V )
17		mpoexga 6236	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
18	16, 16, 17	syl2anc 411	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
19	2, 10, 11, 18	fvmptd3 5629	. 2 |- ( G e. V -> ( +f ` G ) = ( x e. B , y e. B |-> ( x .+ y ) ) )
20	1, 19	eqtrid 2234	1 |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   _Vcvv 2752   Fn wfn 5230   ` cfv 5235  (class class class)co 5895   e. cmpo 5897   Basecbs 12511   +g cplusg 12586   +fcplusf 12826

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-inn 8949  df-ndx 12514  df-slot 12515  df-base 12517  df-plusf 12828

This theorem is referenced by:  plusfvalg  12836  plusfeqg  12837  plusffng  12838  mgmplusf  12839