Theorem isasslaw 41828

Description: The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)

Assertion

Ref	Expression
isasslaw	|- ( ( .o. e. V /\ M e. W ) -> ( .o. assLaw M <-> A. x e. M A. y e. M A. z e. M ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Distinct variable groups:   x, M, y, z   x, .o. , y, z

Allowed substitution hints:   V( x, y, z)   W( x, y, z)

Proof of Theorem isasslaw

Dummy variables m o are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( o = .o. /\ m = M ) -> m = M )
2		id 22	. . . . . . . 8 |- ( o = .o. -> o = .o. )
3		oveq 6656	. . . . . . . 8 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
4		eqidd 2623	. . . . . . . 8 |- ( o = .o. -> z = z )
5	2, 3, 4	oveq123d 6671	. . . . . . 7 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
6		eqidd 2623	. . . . . . . 8 |- ( o = .o. -> x = x )
7		oveq 6656	. . . . . . . 8 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
8	2, 6, 7	oveq123d 6671	. . . . . . 7 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
9	5, 8	eqeq12d 2637	. . . . . 6 |- ( o = .o. -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
10	9	adantr 481	. . . . 5 |- ( ( o = .o. /\ m = M ) -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
11	1, 10	raleqbidv 3152	. . . 4 |- ( ( o = .o. /\ m = M ) -> ( A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. z e. M ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
12	1, 11	raleqbidv 3152	. . 3 |- ( ( o = .o. /\ m = M ) -> ( A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. y e. M A. z e. M ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
13	1, 12	raleqbidv 3152	. 2 |- ( ( o = .o. /\ m = M ) -> ( A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. M A. y e. M A. z e. M ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
14		df-asslaw 41824	. 2 |- assLaw = { <. o , m >. | A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) }
15	13, 14	brabga 4989	1 |- ( ( .o. e. V /\ M e. W ) -> ( .o. assLaw M <-> A. x e. M A. y e. M A. z e. M ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653  (class class class)co 6650   assLaw casslaw 41820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-asslaw 41824

This theorem is referenced by:  asslawass  41829  sgrpplusgaopALT  41831  isassintop  41846  assintopass  41850  sgrp2sgrp  41864