Theorem mnd4g 13294

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 13293	. . 3 |- ( ph -> ( Y .+ ( Z .+ W ) ) = ( Z .+ ( Y .+ W ) ) )
9	8	oveq2d 5962	. 2 |- ( ph -> ( X .+ ( Y .+ ( Z .+ W ) ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
10		mnd4g.2	. . 3 |- ( ph -> X e. B )
11	1, 2	mndcl 13288	. . . 4 |- ( ( G e. Mnd /\ Z e. B /\ W e. B ) -> ( Z .+ W ) e. B )
12	3, 5, 6, 11	syl3anc 1250	. . 3 |- ( ph -> ( Z .+ W ) e. B )
13	1, 2	mndass 13289	. . 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 1252	. 2 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( X .+ ( Y .+ ( Z .+ W ) ) ) )
15	1, 2	mndcl 13288	. . . 4 |- ( ( G e. Mnd /\ Y e. B /\ W e. B ) -> ( Y .+ W ) e. B )
16	3, 4, 6, 15	syl3anc 1250	. . 3 |- ( ph -> ( Y .+ W ) e. B )
17	1, 2	mndass 13289	. . 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 1252	. 2 |- ( ph -> ( ( X .+ Z ) .+ ( Y .+ W ) ) = ( X .+ ( Z .+ ( Y .+ W ) ) ) )
19	9, 14, 18	3eqtr4d 2248	1 |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   ` cfv 5272  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   Mndcmnd 13281

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-inn 9039  df-2 9097  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-mgm 13221  df-sgrp 13267  df-mnd 13282

This theorem is referenced by:  cmn4  13674