Theorem sgrp2sgrp 41864

Description: Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)

Assertion

Ref	Expression
sgrp2sgrp	|- ( M e. SGrpALT <-> M e. SGrp )

Proof of Theorem sgrp2sgrp

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

Step	Hyp	Ref	Expression
1		mgm2mgm 41863	. . . 4 |- ( M e. MgmALT <-> M e. Mgm )
2	1	anbi1i 731	. . 3 |- ( ( M e. MgmALT /\ ( +g ` M ) assLaw ( Base ` M ) ) <-> ( M e. Mgm /\ ( +g ` M ) assLaw ( Base ` M ) ) )
3		fvex 6201	. . . . . 6 |- ( +g ` M ) e. _V
4		fvex 6201	. . . . . 6 |- ( Base ` M ) e. _V
5	3, 4	pm3.2i 471	. . . . 5 |- ( ( +g ` M ) e. _V /\ ( Base ` M ) e. _V )
6		isasslaw 41828	. . . . 5 |- ( ( ( +g ` M ) e. _V /\ ( Base ` M ) e. _V ) -> ( ( +g ` M ) assLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) A. z e. ( Base ` M ) ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) ) )
7	5, 6	mp1i 13	. . . 4 |- ( M e. Mgm -> ( ( +g ` M ) assLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) A. z e. ( Base ` M ) ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) ) )
8	7	pm5.32i 669	. . 3 |- ( ( M e. Mgm /\ ( +g ` M ) assLaw ( Base ` M ) ) <-> ( M e. Mgm /\ A. x e. ( Base ` M ) A. y e. ( Base ` M ) A. z e. ( Base ` M ) ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) ) )
9	2, 8	bitri 264	. 2 |- ( ( M e. MgmALT /\ ( +g ` M ) assLaw ( Base ` M ) ) <-> ( M e. Mgm /\ A. x e. ( Base ` M ) A. y e. ( Base ` M ) A. z e. ( Base ` M ) ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) ) )
10		eqid 2622	. . 3 |- ( Base ` M ) = ( Base ` M )
11		eqid 2622	. . 3 |- ( +g ` M ) = ( +g ` M )
12	10, 11	issgrpALT 41861	. 2 |- ( M e. SGrpALT <-> ( M e. MgmALT /\ ( +g ` M ) assLaw ( Base ` M ) ) )
13	10, 11	issgrp 17285	. 2 |- ( M e. SGrp <-> ( M e. Mgm /\ A. x e. ( Base ` M ) A. y e. ( Base ` M ) A. z e. ( Base ` M ) ( ( x ( +g ` M ) y ) ( +g ` M ) z ) = ( x ( +g ` M ) ( y ( +g ` M ) z ) ) ) )
14	9, 12, 13	3bitr4i 292	1 |- ( M e. SGrpALT <-> M e. SGrp )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   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  SGrpcsgrp 17283   assLaw casslaw 41820  MgmALTcmgm2 41851  SGrpALTcsgrp2 41853

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-sgrp 17284  df-cllaw 41822  df-asslaw 41824  df-mgm2 41855  df-sgrp2 41857

This theorem is referenced by: (None)