Theorem mnd4g 13592

Description: Commutative/associative law for commutative 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 )
mnd4g.5	|- ( ph -> W e. B )
mnd4g.6	|- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )

Assertion

Ref	Expression
mnd4g	|- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Proof of Theorem mnd4g

Step	Hyp	Ref	Expression
1		mndcl.b	. . . 4 |- B = ( Base ` G )
2		mndcl.p	. . . 4 |- .+ = ( +g ` G )
3		mnd4g.1	. . . 4 |- ( ph -> G e. Mnd )
4		mnd4g.3	. . . 4 |- ( ph -> Y e. B )
5		mnd4g.4	. . . 4 |- ( ph -> Z e. B )
6		mnd4g.5	. . . 4 |- ( ph -> W e. B )
7		mnd4g.6	. . . 4 |- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )
8	1, 2, 3, 4, 5, 6, 7	mnd12g 13591	. . 3 |- ( ph -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
9	8	oveq2d 6044	. 2 |- ( ph -> ( X .+ ( Y .+ ( Z .+ W ) ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
10		mnd4g.2	. . 3 |- ( ph -> X e. B )
11	1, 2	mndcl 13586	. . . 4 |- ( ( G e. Mnd /\ Z e. B /\ W e. B ) -> ( Z .+ W ) e. B )
12	3, 5, 6, 11	syl3anc 1274	. . 3 |- ( ph -> ( Z .+ W ) e. B )
13	1, 2	mndass 13587	. . 3 |- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ ( Z .+ W ) e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
14	3, 10, 4, 12, 13	syl13anc 1276	. 2 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
15	1, 2	mndcl 13586	. . . 4 |- ( ( G e. Mnd /\ Y e. B /\ W e. B ) -> ( Y .+ W ) e. B )
16	3, 4, 6, 15	syl3anc 1274	. . 3 |- ( ph -> ( Y .+ W ) e. B )
17	1, 2	mndass 13587	. . 3 |- ( ( G e. Mnd /\ ( X e. B /\ Z e. B /\ ( Y .+ W ) e. B ) ) -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
18	3, 10, 5, 16, 17	syl13anc 1276	. 2 |- ( ph -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
19	9, 14, 18	3eqtr4d 2274	1 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   Mndcmnd 13579

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-mgm 13519  df-sgrp 13565  df-mnd 13580

This theorem is referenced by:  cmn4  13972