Theorem smgrpismgm 10379

Description: A semi-group is a magma. (Contributed by FL, 2-Nov-2009.)

Assertion

Ref	Expression
smgrpismgm	|- (G e. SemiGrp -> G e. Magma)

Proof of Theorem smgrpismgm

Step	Hyp	Ref	Expression
1		df-sgr 10378	. . 3 |- SemiGrp = (Magma i^i Ass)
2	1	eleq2i 1961	. 2 |- (G e. SemiGrp <-> G e. (Magma i^i Ass))
3		elin 2786	. . 3 |- (G e. (Magma i^i Ass) <-> (G e. Magma /\ G e. Ass))
4	3	simplbi 349	. 2 |- (G e. (Magma i^i Ass) -> G e. Magma)
5	2, 4	sylbi 216	1 |- (G e. SemiGrp -> G e. Magma)

Syntax hints:   -> wi 3   e. wcel 1300   i^i cin 2592  Asscass 10359  Magmacmagm 10365  SemiGrpcsem 10377

This theorem is referenced by:  mndismgm 10388  reacomsmgrp2 14704  reacomsmgrp3 14705  resgcom 14712  fprodadd 14713

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-sgr 10378

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