Theorem lmodvneg1 14092

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 14058	. . . . 5 |- ( W e. LMod -> F e. Grp )
4		eqid 2205	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
5		lmodvneg1.u	. . . . . . 7 |- .1. = ( 1r ` F )
6	2, 4, 5	lmod1cl 14077	. . . . . 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 13380	. . . . 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 14067	. . . 4 |- ( ( W e. LMod /\ ( M ` .1. ) e. ( Base ` F ) /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
15	1, 10, 11, 14	syl3anc 1250	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) e. V )
16		eqid 2205	. . . 4 |- ( +g ` W ) = ( +g ` W )
17		eqid 2205	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
18	12, 16, 17	lmod0vrid 14081	. . 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 14090	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )
22	12, 16	lmodass 14065	. . . . 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 1252	. . . 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 14078	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X )
25	24	oveq2d 5960	. . . . . 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 2205	. . . . . . . . . 10 |- ( +g ` F ) = ( +g ` F )
27		eqid 2205	. . . . . . . . . 10 |- ( 0g ` F ) = ( 0g ` F )
28	4, 26, 27, 8	grplinv 13382	. . . . . . . . 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 5959	. . . . . . 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 14074	. . . . . . . 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 1252	. . . . . . 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 14083	. . . . . . 7 |- ( ( W e. LMod /\ X e. V ) -> ( ( 0g ` F ) .x. X ) = ( 0g ` W ) )
34	30, 32, 33	3eqtr3d 2246	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( .1. .x. X ) ) = ( 0g ` W ) )
35	25, 34	eqtr3d 2240	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) X ) = ( 0g ` W ) )
36	35	oveq1d 5959	. . . 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 2240	. . 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 14091	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( X ( +g ` W ) ( N ` X ) ) = ( 0g ` W ) )
39	38	oveq2d 5960	. . 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 14080	. . . 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 2246	. 2 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( M ` .1. ) .x. X ) ( +g ` W ) ( 0g ` W ) ) = ( N ` X ) )
43	19, 42	eqtr3d 2240	1 |- ( ( W e. LMod /\ X e. V ) -> ( ( M ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Scalarcsca 12912   .scvsca 12913   0gc0g 13088   Grpcgrp 13332   invgcminusg 13333   1rcur 13721   LModclmod 14049

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-mgp 13683  df-ur 13722  df-ring 13760  df-lmod 14051

This theorem is referenced by:  lmodvsneg  14093  lmodvsubval2  14104  lssvnegcl  14138  lspsnneg  14182