Theorem plusffvalg 13309

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 13302	. . 3 |- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
3		fveq2 5599	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		plusffval.1	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2258	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5599	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		plusffval.2	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2258	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 5984	. . . 4 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
10	5, 5, 9	mpoeq123dv 6030	. . 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 2788	. . 3 |- ( G e. V -> G e. _V )
12		basfn 13005	. . . . . 6 |- Base Fn _V
13		funfvex 5616	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5395	. . . . . 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 2294	. . . 4 |- ( G e. V -> B e. _V )
17		mpoexga 6321	. . . 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 5696	. 2 |- ( G e. V -> ( +f ` G ) = ( x e. B , y e. B |-> ( x .+ y ) ) )
20	1, 19	eqtrid 2252	1 |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   Basecbs 12947   +g cplusg 13024   +fcplusf 13300

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-plusf 13302

This theorem is referenced by:  plusfvalg  13310  plusfeqg  13311  plusffng  13312  mgmplusf  13313