Theorem mnd12g 13695

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 )
mnd12g.5	|- ( ph -> ( X .+ Y ) = ( Y .+ X ) )

Assertion

Ref	Expression
mnd12g	|- ( ph -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )

Proof of Theorem mnd12g

Step	Hyp	Ref	Expression
1		mnd12g.5	. . 3 |- ( ph -> ( X .+ Y ) = ( Y .+ X ) )
2	1	oveq1d 6075	. 2 |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( Y .+ X ) .+ Z ) )
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 13691	. . 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 1276	. 2 |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
11	7, 8	mndass 13691	. . 3 |- ( ( G e. Mnd /\ ( Y e. B /\ X e. B /\ Z e. B ) ) -> ( ( Y .+ X ) .+ Z ) = ( Y .+ ( X .+ Z ) ) )
12	3, 5, 4, 6, 11	syl13anc 1276	. 2 |- ( ph -> ( ( Y .+ X ) .+ Z ) = ( Y .+ ( X .+ Z ) ) )
13	2, 10, 12	3eqtr3d 2275	1 |- ( ph -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5359  (class class class)co 6060   Basecbs 13302   +g cplusg 13380   Mndcmnd 13683

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-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-cnex 8236  ax-resscn 8237  ax-1re 8239  ax-addrcl 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-iota 5319  df-fun 5361  df-fn 5362  df-fv 5367  df-ov 6063  df-inn 9260  df-2 9318  df-ndx 13305  df-slot 13306  df-base 13308  df-plusg 13393  df-sgrp 13671  df-mnd 13684

This theorem is referenced by:  mnd4g  13696  cmn12  14065