Theorem mgmplusf 17251

Description: The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)

Hypotheses

Ref	Expression
mgmplusf.1	|- B = ( Base ` M )
mgmplusf.2	|- .+^ = ( +f ` M )

Assertion

Ref	Expression
mgmplusf	|- ( M e. Mgm -> .+^ : ( B X. B ) --> B )

Proof of Theorem mgmplusf

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

Step	Hyp	Ref	Expression
1		mgmplusf.1	. . . . 5 |- B = ( Base ` M )
2		eqid 2622	. . . . 5 |- ( +g ` M ) = ( +g ` M )
3	1, 2	mgmcl 17245	. . . 4 |- ( ( M e. Mgm /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
4	3	3expb 1266	. . 3 |- ( ( M e. Mgm /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
5	4	ralrimivva 2971	. 2 |- ( M e. Mgm -> A. x e. B A. y e. B ( x ( +g ` M ) y ) e. B )
6		mgmplusf.2	. . . 4 |- .+^ = ( +f ` M )
7	1, 2, 6	plusffval 17247	. . 3 |- .+^ = ( x e. B , y e. B |-> ( x ( +g ` M ) y ) )
8	7	fmpt2 7237	. 2 |- ( A. x e. B A. y e. B ( x ( +g ` M ) y ) e. B <-> .+^ : ( B X. B ) --> B )
9	5, 8	sylib 208	1 |- ( M e. Mgm -> .+^ : ( B X. B ) --> B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   +fcplusf 17239  Mgmcmgm 17240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-plusf 17241  df-mgm 17242

This theorem is referenced by:  mgmb1mgm1  17254  mndplusf  17309  mgmplusfreseq  41773