Theorem sgrpplusgaopALT 41831

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

Assertion

Ref	Expression
sgrpplusgaopALT	|- ( G e. SGrp -> ( +g ` G ) assLaw ( Base ` G ) )

Proof of Theorem sgrpplusgaopALT

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

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( G e. Mgm /\ A. x e. ( Base ` G ) A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) ) -> A. x e. ( Base ` G ) A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) )
2		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
3		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
4	2, 3	issgrp 17285	. 2 |- ( G e. SGrp <-> ( G e. Mgm /\ A. x e. ( Base ` G ) A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) ) )
5		fvex 6201	. . 3 |- ( +g ` G ) e. _V
6		fvex 6201	. . 3 |- ( Base ` G ) e. _V
7		isasslaw 41828	. . 3 |- ( ( ( +g ` G ) e. _V /\ ( Base ` G ) e. _V ) -> ( ( +g ` G ) assLaw ( Base ` G ) <-> A. x e. ( Base ` G ) A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) ) )
8	5, 6, 7	mp2an 708	. 2 |- ( ( +g ` G ) assLaw ( Base ` G ) <-> A. x e. ( Base ` G ) A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) )
9	1, 4, 8	3imtr4i 281	1 |- ( G e. SGrp -> ( +g ` G ) assLaw ( Base ` G ) )

Syntax hints:   -> wi 4   <-> 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

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-sgrp 17284  df-asslaw 41824

This theorem is referenced by: (None)