Theorem sgrp0 12888

Description: Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)

Assertion

Ref	Expression
sgrp0	|- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Smgrp )

Proof of Theorem sgrp0

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

Step	Hyp	Ref	Expression
1		mgm0 12848	. 2 |- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )
2		rzal 3535	. . 3 |- ( ( 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 ) ) )
3	2	adantl 277	. 2 |- ( ( M e. V /\ ( 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 ) ) )
4		eqid 2189	. . 3 |- ( Base ` M ) = ( Base ` M )
5		eqid 2189	. . 3 |- ( +g ` M ) = ( +g ` M )
6	4, 5	issgrp 12881	. 2 |- ( M e. Smgrp <-> ( 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 ) ) ) )
7	1, 3, 6	sylanbrc 417	1 |- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Smgrp )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   (/)c0 3437   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592  Mgmcmgm 12833  Smgrpcsgrp 12879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5900  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-mgm 12835  df-sgrp 12880

This theorem is referenced by: (None)