Theorem mnd4g 13462

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 13461	. . 3 |- ( ph -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
9	8	oveq2d 6017	. 2 |- ( ph -> ( X .+ ( Y .+ ( Z .+ W ) ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
10		mnd4g.2	. . 3 |- ( ph -> X e. B )
11	1, 2	mndcl 13456	. . . 4 |- ( ( G e. Mnd /\ Z e. B /\ W e. B ) -> ( Z .+ W ) e. B )
12	3, 5, 6, 11	syl3anc 1271	. . 3 |- ( ph -> ( Z .+ W ) e. B )
13	1, 2	mndass 13457	. . 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 1273	. 2 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
15	1, 2	mndcl 13456	. . . 4 |- ( ( G e. Mnd /\ Y e. B /\ W e. B ) -> ( Y .+ W ) e. B )
16	3, 4, 6, 15	syl3anc 1271	. . 3 |- ( ph -> ( Y .+ W ) e. B )
17	1, 2	mndass 13457	. . 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 1273	. 2 |- ( ph -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
19	9, 14, 18	3eqtr4d 2272	1 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   Mndcmnd 13449

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-mgm 13389  df-sgrp 13435  df-mnd 13450

This theorem is referenced by:  cmn4  13842