Theorem sgrpass 13462

Description: A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)

Hypotheses

Ref	Expression
sgrpass.b	|- B = ( Base ` G )
sgrpass.o	|- .o. = ( +g ` G )

Assertion

Ref	Expression
sgrpass	|- ( ( G e. Smgrp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) )

Proof of Theorem sgrpass

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

Step	Hyp	Ref	Expression
1		sgrpass.b	. . . 4 |- B = ( Base ` G )
2		sgrpass.o	. . . 4 |- .o. = ( +g ` G )
3	1, 2	issgrp 13457	. . 3 |- ( G e. Smgrp <-> ( G e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
4		oveq1 6017	. . . . . . 7 |- ( x = X -> ( x .o. y ) = ( X .o. y ) )
5	4	oveq1d 6025	. . . . . 6 |- ( x = X -> ( ( x .o. y ) .o. z ) = ( ( X .o. y ) .o. z ) )
6		oveq1 6017	. . . . . 6 |- ( x = X -> ( x .o. ( y .o. z ) ) = ( X .o. ( y .o. z ) ) )
7	5, 6	eqeq12d 2244	. . . . 5 |- ( x = X -> ( ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) <-> ( ( X .o. y ) .o. z ) = ( X .o. ( y .o. z ) ) ) )
8		oveq2 6018	. . . . . . 7 |- ( y = Y -> ( X .o. y ) = ( X .o. Y ) )
9	8	oveq1d 6025	. . . . . 6 |- ( y = Y -> ( ( X .o. y ) .o. z ) = ( ( X .o. Y ) .o. z ) )
10		oveq1 6017	. . . . . . 7 |- ( y = Y -> ( y .o. z ) = ( Y .o. z ) )
11	10	oveq2d 6026	. . . . . 6 |- ( y = Y -> ( X .o. ( y .o. z ) ) = ( X .o. ( Y .o. z ) ) )
12	9, 11	eqeq12d 2244	. . . . 5 |- ( y = Y -> ( ( ( X .o. y ) .o. z ) = ( X .o. ( y .o. z ) ) <-> ( ( X .o. Y ) .o. z ) = ( X .o. ( Y .o. z ) ) ) )
13		oveq2 6018	. . . . . 6 |- ( z = Z -> ( ( X .o. Y ) .o. z ) = ( ( X .o. Y ) .o. Z ) )
14		oveq2 6018	. . . . . . 7 |- ( z = Z -> ( Y .o. z ) = ( Y .o. Z ) )
15	14	oveq2d 6026	. . . . . 6 |- ( z = Z -> ( X .o. ( Y .o. z ) ) = ( X .o. ( Y .o. Z ) ) )
16	13, 15	eqeq12d 2244	. . . . 5 |- ( z = Z -> ( ( ( X .o. Y ) .o. z ) = ( X .o. ( Y .o. z ) ) <-> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) ) )
17	7, 12, 16	rspc3v 2923	. . . 4 |- ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) ) )
18	17	com12 30	. . 3 |- ( A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) -> ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) ) )
19	3, 18	simplbiim 387	. 2 |- ( G e. Smgrp -> ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) ) )
20	19	imp 124	1 |- ( ( G e. Smgrp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5321  (class class class)co 6010   Basecbs 13053   +g cplusg 13131  Mgmcmgm 13408  Smgrpcsgrp 13455

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-iota 5281  df-fun 5323  df-fn 5324  df-fv 5329  df-ov 6013  df-inn 9127  df-2 9185  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-sgrp 13456

This theorem is referenced by:  prdssgrpd  13469  mndass  13478  dfgrp2  13581  dfgrp3mlem  13652  dfgrp3me  13654  mulgnndir  13709  rngass  13923  rnglidlmsgrp  14482