Theorem mnd32g 17305

Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
mndcl.b	|- B = ( Base ` G )
mndcl.p	|- .+ = ( +g ` G )
mnd4g.1	|- ( ph -> G e. Mnd )
mnd4g.2	|- ( ph -> X e. B )
mnd4g.3	|- ( ph -> Y e. B )
mnd4g.4	|- ( ph -> Z e. B )
mnd32g.5	|- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )

Assertion

Ref	Expression
mnd32g	|- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )

Proof of Theorem mnd32g

Step	Hyp	Ref	Expression
1		mnd32g.5	. . 3 |- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )
2	1	oveq2d 6666	. 2 |- ( ph -> ( X .+ ( Y .+ Z ) ) = ( X .+ ( Z .+ Y ) ) )
3		mnd4g.1	. . 3 |- ( ph -> G e. Mnd )
4		mnd4g.2	. . 3 |- ( ph -> X e. B )
5		mnd4g.3	. . 3 |- ( ph -> Y e. B )
6		mnd4g.4	. . 3 |- ( ph -> Z e. B )
7		mndcl.b	. . . 4 |- B = ( Base ` G )
8		mndcl.p	. . . 4 |- .+ = ( +g ` G )
9	7, 8	mndass 17302	. . 3 |- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
10	3, 4, 5, 6, 9	syl13anc 1328	. 2 |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
11	7, 8	mndass 17302	. . 3 |- ( ( G e. Mnd /\ ( X e. B /\ Z e. B /\ Y e. B ) ) -> ( ( X .+ Z ) .+ Y ) = ( X .+ ( Z .+ Y ) ) )
12	3, 4, 6, 5, 11	syl13anc 1328	. 2 |- ( ph -> ( ( X .+ Z ) .+ Y ) = ( X .+ ( Z .+ Y ) ) )
13	2, 10, 12	3eqtr4d 2666	1 |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294

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-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-iota 5851  df-fv 5896  df-ov 6653  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  cmn32  18211