Theorem sgrpass 12820

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 12815	. . 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 5885	. . . . . . 7 |- ( x = X -> ( x .o. y ) = ( X .o. y ) )
5	4	oveq1d 5893	. . . . . 6 |- ( x = X -> ( ( x .o. y ) .o. z ) = ( ( X .o. y ) .o. z ) )
6		oveq1 5885	. . . . . 6 |- ( x = X -> ( x .o. ( y .o. z ) ) = ( X .o. ( y .o. z ) ) )
7	5, 6	eqeq12d 2192	. . . . 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 5886	. . . . . . 7 |- ( y = Y -> ( X .o. y ) = ( X .o. Y ) )
9	8	oveq1d 5893	. . . . . 6 |- ( y = Y -> ( ( X .o. y ) .o. z ) = ( ( X .o. Y ) .o. z ) )
10		oveq1 5885	. . . . . . 7 |- ( y = Y -> ( y .o. z ) = ( Y .o. z ) )
11	10	oveq2d 5894	. . . . . 6 |- ( y = Y -> ( X .o. ( y .o. z ) ) = ( X .o. ( Y .o. z ) ) )
12	9, 11	eqeq12d 2192	. . . . 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 5886	. . . . . 6 |- ( z = Z -> ( ( X .o. Y ) .o. z ) = ( ( X .o. Y ) .o. Z ) )
14		oveq2 5886	. . . . . . 7 |- ( z = Z -> ( Y .o. z ) = ( Y .o. Z ) )
15	14	oveq2d 5894	. . . . . 6 |- ( z = Z -> ( X .o. ( Y .o. z ) ) = ( X .o. ( Y .o. Z ) ) )
16	13, 15	eqeq12d 2192	. . . . 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 2859	. . . 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5878   Basecbs 12465   +g cplusg 12539  Mgmcmgm 12779  Smgrpcsgrp 12813

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5881  df-inn 8923  df-2 8981  df-ndx 12468  df-slot 12469  df-base 12471  df-plusg 12552  df-sgrp 12814

This theorem is referenced by:  mndass  12831  dfgrp2  12908  dfgrp3mlem  12974  dfgrp3me  12976  mulgnndir  13018