Theorem lmodvneg1 14426

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 14392	. . . . 5 |- ( W e. LMod -> F e. Grp )
4		eqid 2231	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
5		lmodvneg1.u	. . . . . . 7 |- .1. = ( 1r ` F )
6	2, 4, 5	lmod1cl 14411	. . . . . 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 13711	. . . . 5 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( M ` .1. ) e. ( Base ` F ) )
10	3, 7, 9	syl2an2r 599	. . . 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 14401	. . . 4 |- ( ( W e. LMod /\ ( M ` .1. ) e. ( Base ` F ) /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
15	1, 10, 11, 14	syl3anc 1274	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
16		eqid 2231	. . . 4 |- ( +g ` W ) = ( +g ` W )
17		eqid 2231	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
18	12, 16, 17	lmod0vrid 14415	. . 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 14424	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
22	12, 16	lmodass 14399	. . . . 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 1276	. . . 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 14412	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X )
25	24	oveq2d 6044	. . . . . 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 2231	. . . . . . . . . 10 |- ( +g ` F ) = ( +g ` F )
27		eqid 2231	. . . . . . . . . 10 |- ( 0g ` F ) = ( 0g ` F )
28	4, 26, 27, 8	grplinv 13713	. . . . . . . . 9 |- ( ( F e. Grp /\ .1. e. ( Base ` F ) ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
29	3, 7, 28	syl2an2r 599	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) ( +g ` F ) .1. ) = ( 0g ` F ) )
30	29	oveq1d 6043	. . . . . . 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 14408	. . . . . . . 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 1276	. . . . . . 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 14417	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` F ) .x. X ) = ( 0g ` W ) )
34	30, 32, 33	3eqtr3d 2272	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( 0g ` W ) )
35	25, 34	eqtr3d 2266	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) = ( 0g ` W ) )
36	35	oveq1d 6043	. . . 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 2266	. . 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 14425	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( X ( +g ` W ) ( N ` X ) ) = ( 0g ` W ) )
39	38	oveq2d 6044	. . 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 14414	. . . 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 2272	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( N ` X ) )
43	19, 42	eqtr3d 2266	1 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240  Scalarcsca 13243   .scvsca 13244   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664   1rcur 14053   LModclmod 14383

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 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-sca 13256  df-vsca 13257  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-mgp 14015  df-ur 14054  df-ring 14092  df-lmod 14385

This theorem is referenced by:  lmodvsneg  14427  lmodvsubval2  14438  lssvnegcl  14472  lspsnneg  14516