Theorem sgrpass 13427

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 13422	. . 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 6001	. . . . . . 7 |- ( x = X -> ( x .o. y ) = ( X .o. y ) )
5	4	oveq1d 6009	. . . . . 6 |- ( x = X -> ( ( x .o. y ) .o. z ) = ( ( X .o. y ) .o. z ) )
6		oveq1 6001	. . . . . 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 6002	. . . . . . 7 |- ( y = Y -> ( X .o. y ) = ( X .o. Y ) )
9	8	oveq1d 6009	. . . . . 6 |- ( y = Y -> ( ( X .o. y ) .o. z ) = ( ( X .o. Y ) .o. z ) )
10		oveq1 6001	. . . . . . 7 |- ( y = Y -> ( y .o. z ) = ( Y .o. z ) )
11	10	oveq2d 6010	. . . . . 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 6002	. . . . . 6 |- ( z = Z -> ( ( X .o. Y ) .o. z ) = ( ( X .o. Y ) .o. Z ) )
14		oveq2 6002	. . . . . . 7 |- ( z = Z -> ( Y .o. z ) = ( Y .o. Z ) )
15	14	oveq2d 6010	. . . . . 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 5314  (class class class)co 5994   Basecbs 13018   +g cplusg 13096  Mgmcmgm 13373  Smgrpcsgrp 13420

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-ov 5997  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-sgrp 13421

This theorem is referenced by:  prdssgrpd  13434  mndass  13443  dfgrp2  13546  dfgrp3mlem  13617  dfgrp3me  13619  mulgnndir  13674  rngass  13888  rnglidlmsgrp  14446