Theorem sgrpass 12649

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 12644	. . 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 5860	. . . . . . 7 |- ( x = X -> ( x .o. y ) = ( X .o. y ) )
5	4	oveq1d 5868	. . . . . 6 |- ( x = X -> ( ( x .o. y ) .o. z ) = ( ( X .o. y ) .o. z ) )
6		oveq1 5860	. . . . . 6 |- ( x = X -> ( x .o. ( y .o. z ) ) = ( X .o. ( y .o. z ) ) )
7	5, 6	eqeq12d 2185	. . . . 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 5861	. . . . . . 7 |- ( y = Y -> ( X .o. y ) = ( X .o. Y ) )
9	8	oveq1d 5868	. . . . . 6 |- ( y = Y -> ( ( X .o. y ) .o. z ) = ( ( X .o. Y ) .o. z ) )
10		oveq1 5860	. . . . . . 7 |- ( y = Y -> ( y .o. z ) = ( Y .o. z ) )
11	10	oveq2d 5869	. . . . . 6 |- ( y = Y -> ( X .o. ( y .o. z ) ) = ( X .o. ( Y .o. z ) ) )
12	9, 11	eqeq12d 2185	. . . . 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 5861	. . . . . 6 |- ( z = Z -> ( ( X .o. Y ) .o. z ) = ( ( X .o. Y ) .o. Z ) )
14		oveq2 5861	. . . . . . 7 |- ( z = Z -> ( Y .o. z ) = ( Y .o. Z ) )
15	14	oveq2d 5869	. . . . . 6 |- ( z = Z -> ( X .o. ( Y .o. z ) ) = ( X .o. ( Y .o. Z ) ) )
16	13, 15	eqeq12d 2185	. . . . 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 2850	. . . 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 385	. 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 123	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 103   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   ` cfv 5198  (class class class)co 5853   Basecbs 12416   +g cplusg 12480  Mgmcmgm 12608  Smgrpcsgrp 12642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-ov 5856  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-sgrp 12643

This theorem is referenced by:  mndass  12660  dfgrp2  12732