Theorem sgrpass 13051

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 13046	. . 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 5929	. . . . . . 7 |- ( x = X -> ( x .o. y ) = ( X .o. y ) )
5	4	oveq1d 5937	. . . . . 6 |- ( x = X -> ( ( x .o. y ) .o. z ) = ( ( X .o. y ) .o. z ) )
6		oveq1 5929	. . . . . 6 |- ( x = X -> ( x .o. ( y .o. z ) ) = ( X .o. ( y .o. z ) ) )
7	5, 6	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( y = Y -> ( X .o. y ) = ( X .o. Y ) )
9	8	oveq1d 5937	. . . . . 6 |- ( y = Y -> ( ( X .o. y ) .o. z ) = ( ( X .o. Y ) .o. z ) )
10		oveq1 5929	. . . . . . 7 |- ( y = Y -> ( y .o. z ) = ( Y .o. z ) )
11	10	oveq2d 5938	. . . . . 6 |- ( y = Y -> ( X .o. ( y .o. z ) ) = ( X .o. ( Y .o. z ) ) )
12	9, 11	eqeq12d 2211	. . . . 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 5930	. . . . . 6 |- ( z = Z -> ( ( X .o. Y ) .o. z ) = ( ( X .o. Y ) .o. Z ) )
14		oveq2 5930	. . . . . . 7 |- ( z = Z -> ( Y .o. z ) = ( Y .o. Z ) )
15	14	oveq2d 5938	. . . . . 6 |- ( z = Z -> ( X .o. ( Y .o. z ) ) = ( X .o. ( Y .o. Z ) ) )
16	13, 15	eqeq12d 2211	. . . . 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 2884	. . . 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755  Mgmcmgm 12997  Smgrpcsgrp 13044

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-sgrp 13045

This theorem is referenced by:  mndass  13065  dfgrp2  13159  dfgrp3mlem  13230  dfgrp3me  13232  mulgnndir  13281  rngass  13495  rnglidlmsgrp  14053