Theorem baerlem5amN 35373

Description: An equality that holds when X, Y, Z are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 35375 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
baerlem3.v	|- V = ( Base ` W )
baerlem3.m	|- .- = ( -g ` W )
baerlem3.o	|- .0. = ( 0g ` W )
baerlem3.s	|- .(+) = ( LSSum ` W )
baerlem3.n	|- N = ( LSpan ` W )
baerlem3.w	|- ( ph -> W e. LVec )
baerlem3.x	|- ( ph -> X e. V )
baerlem3.c	|- ( ph -> -. X e. ( N ` { Y , Z } ) )
baerlem3.d	|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
baerlem3.y	|- ( ph -> Y e. ( V \ { .0. } ) )
baerlem3.z	|- ( ph -> Z e. ( V \ { .0. } ) )
baerlem5a.p	|- .+ = ( +g ` W )

Assertion

Ref	Expression
baerlem5amN	|- ( ph -> ( N ` { ( X .- ( Y .- Z ) ) } ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) i^i ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) ) )

Proof of Theorem baerlem5amN

Step	Hyp	Ref	Expression
1		baerlem3.y	. . . . . . 7 |- ( ph -> Y e. ( V \ { .0. } ) )
2	1	eldifad 3352	. . . . . 6 |- ( ph -> Y e. V )
3		baerlem3.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
4	3	eldifad 3352	. . . . . 6 |- ( ph -> Z e. V )
5		baerlem3.v	. . . . . . 7 |- V = ( Base ` W )
6		baerlem5a.p	. . . . . . 7 |- .+ = ( +g ` W )
7		eqid 2443	. . . . . . 7 |- ( invg ` W ) = ( invg ` W )
8		baerlem3.m	. . . . . . 7 |- .- = ( -g ` W )
9	5, 6, 7, 8	grpsubval 15593	. . . . . 6 |- ( ( Y e. V /\ Z e. V ) -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
10	2, 4, 9	syl2anc 661	. . . . 5 |- ( ph -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
11	10	oveq2d 6119	. . . 4 |- ( ph -> ( X .- ( Y .- Z ) ) = ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) )
12	11	sneqd 3901	. . 3 |- ( ph -> { ( X .- ( Y .- Z ) ) } = { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } )
13	12	fveq2d 5707	. 2 |- ( ph -> ( N ` { ( X .- ( Y .- Z ) ) } ) = ( N ` { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } ) )
14		baerlem3.o	. . 3 |- .0. = ( 0g ` W )
15		baerlem3.s	. . 3 |- .(+) = ( LSSum ` W )
16		baerlem3.n	. . 3 |- N = ( LSpan ` W )
17		baerlem3.w	. . 3 |- ( ph -> W e. LVec )
18		baerlem3.x	. . 3 |- ( ph -> X e. V )
19		lveclmod 17199	. . . . . 6 |- ( W e. LVec -> W e. LMod )
20	17, 19	syl 16	. . . . 5 |- ( ph -> W e. LMod )
21	5, 7	lmodvnegcl 16998	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( ( invg ` W ) ` Z ) e. V )
22	20, 4, 21	syl2anc 661	. . . 4 |- ( ph -> ( ( invg ` W ) ` Z ) e. V )
23		eqid 2443	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
24	5, 23, 16, 20, 2, 4	lspprcl 17071	. . . . . 6 |- ( ph -> ( N ` { Y , Z } ) e. ( LSubSp ` W ) )
25		baerlem3.c	. . . . . 6 |- ( ph -> -. X e. ( N ` { Y , Z } ) )
26	5, 14, 23, 20, 24, 18, 25	lssneln0 17045	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
27	5, 16, 17, 18, 2, 4, 25	lspindpi 17225	. . . . . 6 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
28	27	simpld 459	. . . . 5 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
29	5, 14, 16, 17, 26, 2, 28	lspsnne1 17210	. . . 4 |- ( ph -> -. X e. ( N ` { Y } ) )
30		baerlem3.d	. . . . . . . 8 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
31	30	necomd 2707	. . . . . . 7 |- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )
32	5, 14, 16, 17, 3, 2, 31	lspsnne1 17210	. . . . . 6 |- ( ph -> -. Z e. ( N ` { Y } ) )
33	5, 16, 17, 18, 4, 2, 32, 25	lspexchn2 17224	. . . . 5 |- ( ph -> -. Z e. ( N ` { Y , X } ) )
34		lmodgrp 16967	. . . . . . . . 9 |- ( W e. LMod -> W e. Grp )
35	17, 19, 34	3syl 20	. . . . . . . 8 |- ( ph -> W e. Grp )
36	35	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. Grp )
37	4	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. V )
38	5, 7	grpinvinv 15605	. . . . . . 7 |- ( ( W e. Grp /\ Z e. V ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
39	36, 37, 38	syl2anc 661	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
40	20	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. LMod )
41	5, 23, 16, 20, 2, 18	lspprcl 17071	. . . . . . . 8 |- ( ph -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
42	41	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
43		simpr 461	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
44	23, 7	lssvnegcl 17049	. . . . . . 7 |- ( ( W e. LMod /\ ( N ` { Y , X } ) e. ( LSubSp ` W ) /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
45	40, 42, 43, 44	syl3anc 1218	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
46	39, 45	eqeltrrd 2518	. . . . 5 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. ( N ` { Y , X } ) )
47	33, 46	mtand 659	. . . 4 |- ( ph -> -. ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
48	5, 16, 17, 22, 18, 2, 29, 47	lspexchn2 17224	. . 3 |- ( ph -> -. X e. ( N ` { Y , ( ( invg ` W ) ` Z ) } ) )
49	5, 7, 16	lspsnneg 17099	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
50	20, 4, 49	syl2anc 661	. . . 4 |- ( ph -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
51	30, 50	neeqtrrd 2644	. . 3 |- ( ph -> ( N ` { Y } ) =/= ( N ` { ( ( invg ` W ) ` Z ) } ) )
52	5, 14, 7	grpinvnzcl 15610	. . . 4 |- ( ( W e. Grp /\ Z e. ( V \ { .0. } ) ) -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
53	35, 3, 52	syl2anc 661	. . 3 |- ( ph -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
54	5, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6	baerlem5a 35371	. 2 |- ( ph -> ( N ` { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) i^i ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) ) )
55	50	oveq2d 6119	. . 3 |- ( ph -> ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) = ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) )
56	5, 6, 8, 7, 35, 18, 4	grpsubinv 15611	. . . . . 6 |- ( ph -> ( X .- ( ( invg ` W ) ` Z ) ) = ( X .+ Z ) )
57	56	sneqd 3901	. . . . 5 |- ( ph -> { ( X .- ( ( invg ` W ) ` Z ) ) } = { ( X .+ Z ) } )
58	57	fveq2d 5707	. . . 4 |- ( ph -> ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) = ( N ` { ( X .+ Z ) } ) )
59	58	oveq1d 6118	. . 3 |- ( ph -> ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) = ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) )
60	55, 59	ineq12d 3565	. 2 |- ( ph -> ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) i^i ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) i^i ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) ) )
61	13, 54, 60	3eqtrd 2479	1 |- ( ph -> ( N ` { ( X .- ( Y .- Z ) ) } ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) i^i ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2618   \ cdif 3337   i^i cin 3339   {csn 3889   {cpr 3891   ` cfv 5430  (class class class)co 6103   Basecbs 14186   +g cplusg 14250   0gc0g 14390   Grpcgrp 15422   invgcminusg 15423   -gcsg 15425   LSSumclsm 16145   LModclmod 16960   LSubSpclss 17025   LSpanclspn 17064   LVecclvec 17195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-cntz 15847  df-lsm 16147  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-drng 16846  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lvec 17196

This theorem is referenced by: (None)