Theorem issmgrpOLD 33662

Description: Obsolete version of issgrp 17285 as of 3-Feb-2020. The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
issmgrpOLD.1	|- X = dom dom G

Assertion

Ref	Expression
issmgrpOLD	|- ( G e. A -> ( G e. SemiGrp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) ) ) )

Distinct variable groups:   x, G, y, z   x, X, y, z

Allowed substitution hints:   A( x, y, z)

Proof of Theorem issmgrpOLD

Step	Hyp	Ref	Expression
1		df-sgrOLD 33660	. . 3 |- SemiGrp = ( Magma i^i Ass )
2	1	eleq2i 2693	. 2 |- ( G e. SemiGrp <-> G e. ( Magma i^i Ass ) )
3		elin 3796	. . 3 |- ( G e. ( Magma i^i Ass ) <-> ( G e. Magma /\ G e. Ass ) )
4		issmgrpOLD.1	. . . . 5 |- X = dom dom G
5	4	ismgmOLD 33649	. . . 4 |- ( G e. A -> ( G e. Magma <-> G : ( X X. X ) --> X ) )
6	4	isass 33645	. . . 4 |- ( G e. A -> ( G e. Ass <-> A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) ) )
7	5, 6	anbi12d 747	. . 3 |- ( G e. A -> ( ( G e. Magma /\ G e. Ass ) <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) ) ) )
8	3, 7	syl5bb 272	. 2 |- ( G e. A -> ( G e. ( Magma i^i Ass ) <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) ) ) )
9	2, 8	syl5bb 272	1 |- ( G e. A -> ( G e. SemiGrp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   X. cxp 5112   dom cdm 5114   -->wf 5884  (class class class)co 6650   Asscass 33641   Magmacmagm 33647   SemiGrpcsem 33659

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-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-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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-ass 33642  df-mgmOLD 33648  df-sgrOLD 33660

This theorem is referenced by:  smgrpmgm  33663  smgrpassOLD  33664  ismndo1  33672