Theorem lmodvneg1 13886

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 109	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> W e. LMod )
2		lmodvneg1.f	. . . . . 6 |- F = (Scalar ` W )
3	2	lmodfgrp 13852	. . . . 5 |- ( W e. LMod -> F e. Grp )
4		eqid 2196	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
5		lmodvneg1.u	. . . . . . 7 |- .1. = ( 1r ` F )
6	2, 4, 5	lmod1cl 13871	. . . . . 6 |- ( W e. LMod -> .1. e. ( Base ` F ) )
7	6	adantr 276	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> .1. e. ( Base ` F ) )
8		lmodvneg1.m	. . . . . 6 |- M = ( invg ` F )
9	4, 8	grpinvcl 13180	. . . . 5 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( M ` .1. ) e. ( Base ` F ) )
10	3, 7, 9	syl2an2r 595	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( M ` .1. ) e. ( Base ` F ) )
11		simpr 110	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> X e. V )
12		lmodvneg1.v	. . . . 5 |- V = ( Base ` W )
13		lmodvneg1.s	. . . . 5 |- .x. = ( .s ` W )
14	12, 2, 13, 4	lmodvscl 13861	. . . 4 |- ( ( W e. LMod /\ ( M ` .1. ) e. ( Base ` F ) /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
15	1, 10, 11, 14	syl3anc 1249	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
16		eqid 2196	. . . 4 |- ( +g ` W ) = ( +g ` W )
17		eqid 2196	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
18	12, 16, 17	lmod0vrid 13875	. . 3 |- ( ( W e. LMod /\ ( ( M ` .1. ) .x. X ) e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( ( M ` .1. ) .x. X ) )
19	15, 18	syldan 282	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( ( M ` .1. ) .x. X ) )
20		lmodvneg1.n	. . . . . 6 |- N = ( invg ` W )
21	12, 20	lmodvnegcl 13884	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
22	12, 16	lmodass 13859	. . . . 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 ) ) ) )
23	1, 15, 11, 21, 22	syl13anc 1251	. . . 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 ) ) ) )
24	12, 2, 13, 5	lmodvs1 13872	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X )
25	24	oveq2d 5938	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) )
26		eqid 2196	. . . . . . . . . 10 |- ( +g ` F ) = ( +g ` F )
27		eqid 2196	. . . . . . . . . 10 |- ( 0g ` F ) = ( 0g ` F )
28	4, 26, 27, 8	grplinv 13182	. . . . . . . . 9 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
29	3, 7, 28	syl2an2r 595	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
30	29	oveq1d 5937	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) ( +g ` F ) .1. ) .x. X ) = ( ( 0g ` F ) .x. X ) )
31	12, 16, 2, 13, 4, 26	lmodvsdir 13868	. . . . . . . 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 ) ) )
32	1, 10, 7, 11, 31	syl13anc 1251	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) ( +g ` F ) .1. ) .x. X ) = ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) )
33	12, 2, 13, 27, 17	lmod0vs 13877	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` F ) .x. X ) = ( 0g ` W ) )
34	30, 32, 33	3eqtr3d 2237	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( 0g ` W ) )
35	25, 34	eqtr3d 2231	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) = ( 0g ` W ) )
36	35	oveq1d 5937	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) ( +g ` W ) ( N ` X ) ) = ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) )
37	23, 36	eqtr3d 2231	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( X ( +g ` W ) ( N ` X ) ) ) = ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) )
38	12, 16, 17, 20	lmodvnegid 13885	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( X ( +g ` W ) ( N ` X ) ) = ( 0g ` W ) )
39	38	oveq2d 5938	. . 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 ) ) )
40	12, 16, 17	lmod0vlid 13874	. . . 4 |- ( ( W e. LMod /\ ( N ` X ) e. V ) -> ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) = ( N ` X ) )
41	21, 40	syldan 282	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` W ) ( +g ` W ) ( N ` X ) ) = ( N ` X ) )
42	37, 39, 41	3eqtr3d 2237	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( N ` X ) )
43	19, 42	eqtr3d 2231	1 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755  Scalarcsca 12758   .scvsca 12759   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133   1rcur 13515   LModclmod 13843

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mgp 13477  df-ur 13516  df-ring 13554  df-lmod 13845

This theorem is referenced by:  lmodvsneg  13887  lmodvsubval2  13898  lssvnegcl  13932  lspsnneg  13976