Theorem mgmplusgiopALT 41830

Description: Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
mgmplusgiopALT	|- ( M e. Mgm -> ( +g ` M ) clLaw ( Base ` M ) )

Proof of Theorem mgmplusgiopALT

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. . . . 5 |- ( +g ` M ) = ( +g ` M )
3	1, 2	mgmcl 17245	. . . 4 |- ( ( M e. Mgm /\ x e. ( Base ` M ) /\ y e. ( Base ` M ) ) -> ( x ( +g ` M ) y ) e. ( Base ` M ) )
4	3	3expb 1266	. . 3 |- ( ( M e. Mgm /\ ( x e. ( Base ` M ) /\ y e. ( Base ` M ) ) ) -> ( x ( +g ` M ) y ) e. ( Base ` M ) )
5	4	ralrimivva 2971	. 2 |- ( M e. Mgm -> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
6		fvex 6201	. . . 4 |- ( +g ` M ) e. _V
7		fvex 6201	. . . 4 |- ( Base ` M ) e. _V
8	6, 7	pm3.2i 471	. . 3 |- ( ( +g ` M ) e. _V /\ ( Base ` M ) e. _V )
9		iscllaw 41825	. . 3 |- ( ( ( +g ` M ) e. _V /\ ( Base ` M ) e. _V ) -> ( ( +g ` M ) clLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
10	8, 9	mp1i 13	. 2 |- ( M e. Mgm -> ( ( +g ` M ) clLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
11	5, 10	mpbird 247	1 |- ( M e. Mgm -> ( +g ` M ) clLaw ( Base ` M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240   clLaw ccllaw 41819

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-cllaw 41822

This theorem is referenced by:  mgm2mgm  41863