Theorem lmodvneg1 18906

Description: Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lmodvneg1.v	|- V = ( Base ` W )
lmodvneg1.n	|- N = ( invg ` W )
lmodvneg1.f	|- F = (Scalar ` W )
lmodvneg1.s	|- .x. = ( .s ` W )
lmodvneg1.u	|- .1. = ( 1r ` F )
lmodvneg1.m	|- M = ( invg ` F )

Assertion

Ref	Expression
lmodvneg1	|- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )

Proof of Theorem lmodvneg1

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> W e. LMod )
2		lmodvneg1.f	. . . . . . 7 |- F = (Scalar ` W )
3	2	lmodfgrp 18872	. . . . . 6 |- ( W e. LMod -> F e. Grp )
4	3	adantr 481	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> F e. Grp )
5		eqid 2622	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmodvneg1.u	. . . . . . 7 |- .1. = ( 1r ` F )
7	2, 5, 6	lmod1cl 18890	. . . . . 6 |- ( W e. LMod -> .1. e. ( Base ` F ) )
8	7	adantr 481	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> .1. e. ( Base ` F ) )
9		lmodvneg1.m	. . . . . 6 |- M = ( invg ` F )
10	5, 9	grpinvcl 17467	. . . . 5 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( M ` .1. ) e. ( Base ` F ) )
11	4, 8, 10	syl2anc 693	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( M ` .1. ) e. ( Base ` F ) )
12		simpr 477	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> X e. V )
13		lmodvneg1.v	. . . . 5 |- V = ( Base ` W )
14		lmodvneg1.s	. . . . 5 |- .x. = ( .s ` W )
15	13, 2, 14, 5	lmodvscl 18880	. . . 4 |- ( ( W e. LMod /\ ( M ` .1. ) e. ( Base ` F ) /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
16	1, 11, 12, 15	syl3anc 1326	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
17		eqid 2622	. . . 4 |- ( +g ` W ) = ( +g ` W )
18		eqid 2622	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
19	13, 17, 18	lmod0vrid 18894	. . 3 |- ( ( W e. LMod /\ ( ( M ` .1. ) .x. X ) e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( ( M ` .1. ) .x. X ) )
20	16, 19	syldan 487	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( ( M ` .1. ) .x. X ) )
21		lmodvneg1.n	. . . . . 6 |- N = ( invg ` W )
22	13, 21	lmodvnegcl 18904	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
23	13, 17	lmodass 18878	. . . . 5 |- ( ( W e. LMod /\ ( ( ( M ` .1. ) .x. X ) e. V /\ X e. V /\ ( N ` X ) e. V ) ) -> ( ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) ( +g ` W ) ( N ` X ) ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( X ( +g ` W ) ( N ` X ) ) ) )
24	1, 16, 12, 22, 23	syl13anc 1328	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) ( +g ` W ) ( N ` X ) ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( X ( +g ` W ) ( N ` X ) ) ) )
25	13, 2, 14, 6	lmodvs1 18891	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X )
26	25	oveq2d 6666	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) )
27		eqid 2622	. . . . . . . . . 10 |- ( +g ` F ) = ( +g ` F )
28		eqid 2622	. . . . . . . . . 10 |- ( 0g ` F ) = ( 0g ` F )
29	5, 27, 28, 9	grplinv 17468	. . . . . . . . 9 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
30	4, 8, 29	syl2anc 693	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
31	30	oveq1d 6665	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) ( +g ` F ) .1. ) .x. X ) = ( ( 0g ` F ) .x. X ) )
32	13, 17, 2, 14, 5, 27	lmodvsdir 18887	. . . . . . . 8 |- ( ( W e. LMod /\ ( ( M ` .1. ) e. ( Base ` F ) /\ .1. e. ( Base ` F ) /\ X e. V ) ) -> ( ( ( M ` .1. ) ( +g ` F ) .1. ) .x. X ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) )
33	1, 11, 8, 12, 32	syl13anc 1328	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) ( +g ` F ) .1. ) .x. X ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) )
34	13, 2, 14, 28, 18	lmod0vs 18896	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` F ) .x. X ) = ( 0g ` W ) )
35	31, 33, 34	3eqtr3d 2664	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( 0g ` W ) )
36	26, 35	eqtr3d 2658	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) = ( 0g ` W ) )
37	36	oveq1d 6665	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) ( +g ` W ) ( N ` X ) ) = ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) )
38	24, 37	eqtr3d 2658	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( X ( +g ` W ) ( N ` X ) ) ) = ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) )
39	13, 17, 18, 21	lmodvnegid 18905	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( X ( +g ` W ) ( N ` X ) ) = ( 0g ` W ) )
40	39	oveq2d 6666	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( X ( +g ` W ) ( N ` X ) ) ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) )
41	13, 17, 18	lmod0vlid 18893	. . . 4 |- ( ( W e. LMod /\ ( N ` X ) e. V ) -> ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) = ( N ` X ) )
42	22, 41	syldan 487	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) = ( N ` X ) )
43	38, 40, 42	3eqtr3d 2664	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( N ` X ) )
44	20, 43	eqtr3d 2658	1 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   1rcur 18501   LModclmod 18863

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865

This theorem is referenced by:  lmodvsneg  18907  lmodvsubval2  18918  lssvnegcl  18956  lspsnneg  19006  lmodvsinv  19036  lspsolvlem  19142  tlmtgp  21999  clmvneg1  22899  deg1invg  23866