Theorem issmgrp 10381

Description: The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.)

Hypothesis

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

Assertion

Ref	Expression
issmgrp	|- (G e. A -> (G e. SemiGrp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)))))

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

Proof of Theorem issmgrp

Step	Hyp	Ref	Expression
1		issmgrp.1	. . . . 5 |- X = dom dom G
2	1	ismgm 10367	. . . 4 |- (G e. A -> (G e. Magma <-> G:(X X. X)-->X))
3	1	isass 10363	. . . 4 |- (G e. A -> (G e. Ass <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
4	2, 3	anbi12d 690	. . 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 ((xGy)Gz) = (xG(yGz)))))
5		elin 2786	. . 3 |- (G e. (Magma i^i Ass) <-> (G e. Magma /\ G e. Ass))
6	4, 5	syl5bb 591	. 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 ((xGy)Gz) = (xG(yGz)))))
7		df-sgr 10378	. . 3 |- SemiGrp = (Magma i^i Ass)
8	7	eleq2i 1961	. 2 |- (G e. SemiGrp <-> G e. (Magma i^i Ass))
9	6, 8	syl5bb 591	1 |- (G e. A -> (G e. SemiGrp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592   X. cxp 3984  dom cdm 3986  -->wf 3994  (class class class)co 4884  Asscass 10359  Magmacmagm 10365  SemiGrpcsem 10377

This theorem is referenced by:  smgrpmgm 10382  smgrpass 10383  ismnd1 10391

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-ass 10360  df-mgm 10366  df-sgr 10378

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