Theorem cmn4 18212

Description: Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
ablcom.b	|- B = ( Base ` G )
ablcom.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
cmn4	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Proof of Theorem cmn4

Step	Hyp	Ref	Expression
1		ablcom.b	. 2 |- B = ( Base ` G )
2		ablcom.p	. 2 |- .+ = ( +g ` G )
3		simp1 1061	. . 3 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> G e. CMnd )
4		cmnmnd 18208	. . 3 |- ( G e. CMnd -> G e. Mnd )
5	3, 4	syl 17	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> G e. Mnd )
6		simp2l 1087	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> X e. B )
7		simp2r 1088	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Y e. B )
8		simp3l 1089	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Z e. B )
9		simp3r 1090	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> W e. B )
10	1, 2	cmncom 18209	. . 3 |- ( ( G e. CMnd /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
11	3, 7, 8, 10	syl3anc 1326	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
12	1, 2, 5, 6, 7, 8, 9, 11	mnd4g 17307	1 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195

This theorem is referenced by:  ablsub4  18218  ghmplusg  18249  lmod4  18913  evlslem1  19515  ip2di  19986  clmsub4  22906  lfladdcl  34358