Theorem mnd4g 13013

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 13012	. . 3 |- ( ph -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
9	8	oveq2d 5935	. 2 |- ( ph -> ( X .+ ( Y .+ ( Z .+ W ) ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
10		mnd4g.2	. . 3 |- ( ph -> X e. B )
11	1, 2	mndcl 13007	. . . 4 |- ( ( G e. Mnd /\ Z e. B /\ W e. B ) -> ( Z .+ W ) e. B )
12	3, 5, 6, 11	syl3anc 1249	. . 3 |- ( ph -> ( Z .+ W ) e. B )
13	1, 2	mndass 13008	. . 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 1251	. 2 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
15	1, 2	mndcl 13007	. . . 4 |- ( ( G e. Mnd /\ Y e. B /\ W e. B ) -> ( Y .+ W ) e. B )
16	3, 4, 6, 15	syl3anc 1249	. . 3 |- ( ph -> ( Y .+ W ) e. B )
17	1, 2	mndass 13008	. . 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 1251	. 2 |- ( ph -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
19	9, 14, 18	3eqtr4d 2236	1 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   Mndcmnd 13000

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mgm 12942  df-sgrp 12988  df-mnd 13001

This theorem is referenced by:  cmn4  13378