Theorem lmodcom 14066

Description: Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)

Hypotheses

Ref	Expression
lmodcom.v	|- V = ( Base ` W )
lmodcom.a	|- .+ = ( +g ` W )

Assertion

Ref	Expression
lmodcom	|- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) = ( Y .+ X ) )

Proof of Theorem lmodcom

Step	Hyp	Ref	Expression
1		simp1 999	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> W e. LMod )
2		eqid 2204	. . . . . . . . . . 11 |- (Scalar ` W ) = (Scalar ` W )
3		eqid 2204	. . . . . . . . . . 11 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
4		eqid 2204	. . . . . . . . . . 11 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
5	2, 3, 4	lmod1cl 14048	. . . . . . . . . 10 |- ( W e. LMod -> ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
6	1, 5	syl 14	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
7		eqid 2204	. . . . . . . . . 10 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
8	2, 3, 7	lmodacl 14032	. . . . . . . . 9 |- ( ( W e. LMod /\ ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) ) -> ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) e. ( Base ` (Scalar ` W ) ) )
9	1, 6, 6, 8	syl3anc 1249	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) e. ( Base ` (Scalar ` W ) ) )
10		simp2 1000	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> X e. V )
11		simp3 1001	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> Y e. V )
12		lmodcom.v	. . . . . . . . 9 |- V = ( Base ` W )
13		lmodcom.a	. . . . . . . . 9 |- .+ = ( +g ` W )
14		eqid 2204	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
15	12, 13, 2, 14, 3	lmodvsdi 14044	. . . . . . . 8 |- ( ( W e. LMod /\ ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) e. ( Base ` (Scalar ` W ) ) /\ X e. V /\ Y e. V ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) ( X .+ Y ) ) = ( ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) .+ ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) ) )
16	1, 9, 10, 11, 15	syl13anc 1251	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) ( X .+ Y ) ) = ( ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) .+ ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) ) )
17	12, 13	lmodvacl 14035	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
18	12, 13, 2, 14, 3, 7	lmodvsdir 14045	. . . . . . . 8 |- ( ( W e. LMod /\ ( ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ ( X .+ Y ) e. V ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) ( X .+ Y ) ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) ) )
19	1, 6, 6, 17, 18	syl13anc 1251	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) ( X .+ Y ) ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) ) )
20	16, 19	eqtr3d 2239	. . . . . 6 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) .+ ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) ) )
21	12, 13, 2, 14, 3, 7	lmodvsdir 14045	. . . . . . . . 9 |- ( ( W e. LMod /\ ( ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ X e. V ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) ) )
22	1, 6, 6, 10, 21	syl13anc 1251	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) ) )
23	12, 2, 14, 4	lmodvs1 14049	. . . . . . . . . 10 |- ( ( W e. LMod /\ X e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) = X )
24	1, 10, 23	syl2anc 411	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) = X )
25	24, 24	oveq12d 5961	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) ) = ( X .+ X ) )
26	22, 25	eqtrd 2237	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) = ( X .+ X ) )
27	12, 13, 2, 14, 3, 7	lmodvsdir 14045	. . . . . . . . 9 |- ( ( W e. LMod /\ ( ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) /\ Y e. V ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) ) )
28	1, 6, 6, 11, 27	syl13anc 1251	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) = ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) ) )
29	12, 2, 14, 4	lmodvs1 14049	. . . . . . . . . 10 |- ( ( W e. LMod /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) = Y )
30	1, 11, 29	syl2anc 411	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) = Y )
31	30, 30	oveq12d 5961	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) ) = ( Y .+ Y ) )
32	28, 31	eqtrd 2237	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) = ( Y .+ Y ) )
33	26, 32	oveq12d 5961	. . . . . 6 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) X ) .+ ( ( ( 1r ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 1r ` (Scalar ` W ) ) ) ( .s ` W ) Y ) ) = ( ( X .+ X ) .+ ( Y .+ Y ) ) )
34	12, 2, 14, 4	lmodvs1 14049	. . . . . . . 8 |- ( ( W e. LMod /\ ( X .+ Y ) e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) = ( X .+ Y ) )
35	1, 17, 34	syl2anc 411	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) = ( X .+ Y ) )
36	35, 35	oveq12d 5961	. . . . . 6 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) .+ ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) ( X .+ Y ) ) ) = ( ( X .+ Y ) .+ ( X .+ Y ) ) )
37	20, 33, 36	3eqtr3d 2245	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ X ) .+ ( Y .+ Y ) ) = ( ( X .+ Y ) .+ ( X .+ Y ) ) )
38	12, 13	lmodvacl 14035	. . . . . . 7 |- ( ( W e. LMod /\ X e. V /\ X e. V ) -> ( X .+ X ) e. V )
39	1, 10, 10, 38	syl3anc 1249	. . . . . 6 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ X ) e. V )
40	12, 13	lmodass 14036	. . . . . 6 |- ( ( W e. LMod /\ ( ( X .+ X ) e. V /\ Y e. V /\ Y e. V ) ) -> ( ( ( X .+ X ) .+ Y ) .+ Y ) = ( ( X .+ X ) .+ ( Y .+ Y ) ) )
41	1, 39, 11, 11, 40	syl13anc 1251	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( X .+ X ) .+ Y ) .+ Y ) = ( ( X .+ X ) .+ ( Y .+ Y ) ) )
42	12, 13	lmodass 14036	. . . . . 6 |- ( ( W e. LMod /\ ( ( X .+ Y ) e. V /\ X e. V /\ Y e. V ) ) -> ( ( ( X .+ Y ) .+ X ) .+ Y ) = ( ( X .+ Y ) .+ ( X .+ Y ) ) )
43	1, 17, 10, 11, 42	syl13anc 1251	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( X .+ Y ) .+ X ) .+ Y ) = ( ( X .+ Y ) .+ ( X .+ Y ) ) )
44	37, 41, 43	3eqtr4d 2247	. . . 4 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( X .+ X ) .+ Y ) .+ Y ) = ( ( ( X .+ Y ) .+ X ) .+ Y ) )
45		lmodgrp 14027	. . . . . 6 |- ( W e. LMod -> W e. Grp )
46	1, 45	syl 14	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> W e. Grp )
47	12, 13	lmodvacl 14035	. . . . . 6 |- ( ( W e. LMod /\ ( X .+ X ) e. V /\ Y e. V ) -> ( ( X .+ X ) .+ Y ) e. V )
48	1, 39, 11, 47	syl3anc 1249	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ X ) .+ Y ) e. V )
49	12, 13	lmodvacl 14035	. . . . . 6 |- ( ( W e. LMod /\ ( X .+ Y ) e. V /\ X e. V ) -> ( ( X .+ Y ) .+ X ) e. V )
50	1, 17, 10, 49	syl3anc 1249	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ Y ) .+ X ) e. V )
51	12, 13	grprcan 13340	. . . . 5 |- ( ( W e. Grp /\ ( ( ( X .+ X ) .+ Y ) e. V /\ ( ( X .+ Y ) .+ X ) e. V /\ Y e. V ) ) -> ( ( ( ( X .+ X ) .+ Y ) .+ Y ) = ( ( ( X .+ Y ) .+ X ) .+ Y ) <-> ( ( X .+ X ) .+ Y ) = ( ( X .+ Y ) .+ X ) ) )
52	46, 48, 50, 11, 51	syl13anc 1251	. . . 4 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( ( ( X .+ X ) .+ Y ) .+ Y ) = ( ( ( X .+ Y ) .+ X ) .+ Y ) <-> ( ( X .+ X ) .+ Y ) = ( ( X .+ Y ) .+ X ) ) )
53	44, 52	mpbid 147	. . 3 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ X ) .+ Y ) = ( ( X .+ Y ) .+ X ) )
54	12, 13	lmodass 14036	. . . 4 |- ( ( W e. LMod /\ ( X e. V /\ X e. V /\ Y e. V ) ) -> ( ( X .+ X ) .+ Y ) = ( X .+ ( X .+ Y ) ) )
55	1, 10, 10, 11, 54	syl13anc 1251	. . 3 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ X ) .+ Y ) = ( X .+ ( X .+ Y ) ) )
56	12, 13	lmodass 14036	. . . 4 |- ( ( W e. LMod /\ ( X e. V /\ Y e. V /\ X e. V ) ) -> ( ( X .+ Y ) .+ X ) = ( X .+ ( Y .+ X ) ) )
57	1, 10, 11, 10, 56	syl13anc 1251	. . 3 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ Y ) .+ X ) = ( X .+ ( Y .+ X ) ) )
58	53, 55, 57	3eqtr3d 2245	. 2 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ ( X .+ Y ) ) = ( X .+ ( Y .+ X ) ) )
59	12, 13	lmodvacl 14035	. . . 4 |- ( ( W e. LMod /\ Y e. V /\ X e. V ) -> ( Y .+ X ) e. V )
60	59	3com23 1211	. . 3 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( Y .+ X ) e. V )
61	12, 13	lmodlcan 14037	. . 3 |- ( ( W e. LMod /\ ( ( X .+ Y ) e. V /\ ( Y .+ X ) e. V /\ X e. V ) ) -> ( ( X .+ ( X .+ Y ) ) = ( X .+ ( Y .+ X ) ) <-> ( X .+ Y ) = ( Y .+ X ) ) )
62	1, 17, 60, 10, 61	syl13anc 1251	. 2 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .+ ( X .+ Y ) ) = ( X .+ ( Y .+ X ) ) <-> ( X .+ Y ) = ( Y .+ X ) ) )
63	58, 62	mpbid 147	1 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) = ( Y .+ X ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880  Scalarcsca 12883   .scvsca 12884   Grpcgrp 13303   1rcur 13692   LModclmod 14020

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-plusg 12893  df-mulr 12894  df-sca 12896  df-vsca 12897  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-mgp 13654  df-ur 13693  df-ring 13731  df-lmod 14022

This theorem is referenced by:  lmodabl  14067  lssvsubcl  14099  lssvancl2  14101