Theorem plusffvalg 12616

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 12609	. . 3 |- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
3		fveq2 5496	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		plusffval.1	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2221	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5496	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		plusffval.2	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2221	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 5870	. . . 4 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
10	5, 5, 9	mpoeq123dv 5915	. . 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 2741	. . 3 |- ( G e. V -> G e. _V )
12		basfn 12473	. . . . . 6 |- Base Fn _V
13		funfvex 5513	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5298	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
15	12, 11, 14	sylancr 412	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
16	4, 15	eqeltrid 2257	. . . 4 |- ( G e. V -> B e. _V )
17		mpoexga 6191	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
18	16, 16, 17	syl2anc 409	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ y ) ) e. _V )
19	2, 10, 11, 18	fvmptd3 5589	. 2 |- ( G e. V -> ( +f ` G ) = ( x e. B , y e. B |-> ( x .+ y ) ) )
20	1, 19	eqtrid 2215	1 |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   Basecbs 12416   +g cplusg 12480   +fcplusf 12607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-plusf 12609

This theorem is referenced by:  plusfvalg  12617  plusfeqg  12618  plusffng  12619  mgmplusf  12620