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Theorem issgrpv 15891
Description: The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
Hypotheses
Ref Expression
issgrpn0.b  |-  B  =  ( Base `  M
)
issgrpn0.o  |-  .o.  =  ( +g  `  M )
Assertion
Ref Expression
issgrpv  |-  ( M  e.  V  ->  ( M  e. SGrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y )  .o.  z
)  =  ( x  .o.  ( y  .o.  z ) ) ) ) )
Distinct variable groups:    x, B, y, z    x, M, y, z    x,  .o. , y, z
Allowed substitution hints:    V( x, y, z)

Proof of Theorem issgrpv
StepHypRef Expression
1 issgrpn0.b . . . 4  |-  B  =  ( Base `  M
)
2 issgrpn0.o . . . 4  |-  .o.  =  ( +g  `  M )
31, 2ismgm 15851 . . 3  |-  ( M  e.  V  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y
)  e.  B ) )
43anbi1d 704 . 2  |-  ( M  e.  V  ->  (
( M  e. Mgm  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) )  <->  ( A. x  e.  B  A. y  e.  B  (
x  .o.  y )  e.  B  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) ) )
51, 2issgrp 15890 . 2  |-  ( M  e. SGrp 
<->  ( M  e. Mgm  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) ) )
6 r19.26-2 2971 . 2  |-  ( A. x  e.  B  A. y  e.  B  (
( x  .o.  y
)  e.  B  /\  A. z  e.  B  ( ( x  .o.  y
)  .o.  z )  =  ( x  .o.  ( y  .o.  z
) ) )  <->  ( A. x  e.  B  A. y  e.  B  (
x  .o.  y )  e.  B  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) )
74, 5, 63bitr4g 288 1  |-  ( M  e.  V  ->  ( M  e. SGrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y )  .o.  z
)  =  ( x  .o.  ( y  .o.  z ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   ` cfv 5578  (class class class)co 6281   Basecbs 14613   +g cplusg 14678  Mgmcmgm 15848  SGrpcsgrp 15888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-mgm 15850  df-sgrp 15889
This theorem is referenced by:  ismnd  15901
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