Theorem baerlem5amN 32199

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 32201 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 3292	. . . . . 6 |- ( ph -> Y e. V )
3		baerlem3.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
4	3	eldifad 3292	. . . . . 6 |- ( ph -> Z e. V )
5		baerlem3.v	. . . . . . 7 |- V = ( Base ` W )
6		baerlem5a.p	. . . . . . 7 |- .+ = ( +g ` W )
7		eqid 2404	. . . . . . 7 |- ( inv g ` W ) = ( inv g ` W )
8		baerlem3.m	. . . . . . 7 |- .- = ( -g ` W )
9	5, 6, 7, 8	grpsubval 14803	. . . . . 6 |- ( ( Y e. V /\ Z e. V ) -> ( Y .- Z ) = ( Y .+ ( ( inv g ` W ) ` Z ) ) )
10	2, 4, 9	syl2anc 643	. . . . 5 |- ( ph -> ( Y .- Z ) = ( Y .+ ( ( inv g ` W ) ` Z ) ) )
11	10	oveq2d 6056	. . . 4 |- ( ph -> ( X .- ( Y .- Z ) ) = ( X .- ( Y .+ ( ( inv g ` W ) ` Z ) ) ) )
12	11	sneqd 3787	. . 3 |- ( ph -> { ( X .- ( Y .- Z ) ) } = { ( X .- ( Y .+ ( ( inv g ` W ) ` Z ) ) ) } )
13	12	fveq2d 5691	. 2 |- ( ph -> ( N ` { ( X .- ( Y .- Z ) ) } ) = ( N ` { ( X .- ( Y .+ ( ( inv g ` 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 16133	. . . . . 6 |- ( W e. LVec -> W e. LMod )
20	17, 19	syl 16	. . . . 5 |- ( ph -> W e. LMod )
21	5, 7	lmodvnegcl 15940	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( ( inv g ` W ) ` Z ) e. V )
22	20, 4, 21	syl2anc 643	. . . 4 |- ( ph -> ( ( inv g ` W ) ` Z ) e. V )
23		eqid 2404	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
24	5, 23, 16, 20, 2, 4	lspprcl 16009	. . . . . 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 15983	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
27	5, 16, 17, 18, 2, 4, 25	lspindpi 16159	. . . . . 6 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
28	27	simpld 446	. . . . 5 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
29	5, 14, 16, 17, 26, 2, 28	lspsnne1 16144	. . . 4 |- ( ph -> -. X e. ( N ` { Y } ) )
30		baerlem3.d	. . . . . . . 8 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
31	30	necomd 2650	. . . . . . 7 |- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )
32	5, 14, 16, 17, 3, 2, 31	lspsnne1 16144	. . . . . 6 |- ( ph -> -. Z e. ( N ` { Y } ) )
33	5, 16, 17, 18, 4, 2, 32, 25	lspexchn2 16158	. . . . 5 |- ( ph -> -. Z e. ( N ` { Y , X } ) )
34		lmodgrp 15912	. . . . . . . . 9 |- ( W e. LMod -> W e. Grp )
35	17, 19, 34	3syl 19	. . . . . . . 8 |- ( ph -> W e. Grp )
36	35	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. Grp )
37	4	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. V )
38	5, 7	grpinvinv 14813	. . . . . . 7 |- ( ( W e. Grp /\ Z e. V ) -> ( ( inv g ` W ) ` ( ( inv g ` W ) ` Z ) ) = Z )
39	36, 37, 38	syl2anc 643	. . . . . 6 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( inv g ` W ) ` ( ( inv g ` W ) ` Z ) ) = Z )
40	20	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. LMod )
41	5, 23, 16, 20, 2, 18	lspprcl 16009	. . . . . . . 8 |- ( ph -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
42	41	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
43		simpr 448	. . . . . . 7 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) )
44	23, 7	lssvnegcl 15987	. . . . . . 7 |- ( ( W e. LMod /\ ( N ` { Y , X } ) e. ( LSubSp ` W ) /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( inv g ` W ) ` ( ( inv g ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
45	40, 42, 43, 44	syl3anc 1184	. . . . . 6 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( inv g ` W ) ` ( ( inv g ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
46	39, 45	eqeltrrd 2479	. . . . 5 |- ( ( ph /\ ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. ( N ` { Y , X } ) )
47	33, 46	mtand 641	. . . 4 |- ( ph -> -. ( ( inv g ` W ) ` Z ) e. ( N ` { Y , X } ) )
48	5, 16, 17, 22, 18, 2, 29, 47	lspexchn2 16158	. . 3 |- ( ph -> -. X e. ( N ` { Y , ( ( inv g ` W ) ` Z ) } ) )
49	5, 7, 16	lspsnneg 16037	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( N ` { ( ( inv g ` W ) ` Z ) } ) = ( N ` { Z } ) )
50	20, 4, 49	syl2anc 643	. . . 4 |- ( ph -> ( N ` { ( ( inv g ` W ) ` Z ) } ) = ( N ` { Z } ) )
51	30, 50	neeqtrrd 2591	. . 3 |- ( ph -> ( N ` { Y } ) =/= ( N ` { ( ( inv g ` W ) ` Z ) } ) )
52	5, 14, 7	grpinvnzcl 14818	. . . 4 |- ( ( W e. Grp /\ Z e. ( V \ { .0. } ) ) -> ( ( inv g ` W ) ` Z ) e. ( V \ { .0. } ) )
53	35, 3, 52	syl2anc 643	. . 3 |- ( ph -> ( ( inv g ` W ) ` Z ) e. ( V \ { .0. } ) )
54	5, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6	baerlem5a 32197	. 2 |- ( ph -> ( N ` { ( X .- ( Y .+ ( ( inv g ` W ) ` Z ) ) ) } ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( inv g ` W ) ` Z ) } ) ) i^i ( ( N ` { ( X .- ( ( inv g ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) ) )
55	50	oveq2d 6056	. . 3 |- ( ph -> ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( inv g ` W ) ` Z ) } ) ) = ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) )
56	5, 6, 8, 7, 35, 18, 4	grpsubinv 14819	. . . . . 6 |- ( ph -> ( X .- ( ( inv g ` W ) ` Z ) ) = ( X .+ Z ) )
57	56	sneqd 3787	. . . . 5 |- ( ph -> { ( X .- ( ( inv g ` W ) ` Z ) ) } = { ( X .+ Z ) } )
58	57	fveq2d 5691	. . . 4 |- ( ph -> ( N ` { ( X .- ( ( inv g ` W ) ` Z ) ) } ) = ( N ` { ( X .+ Z ) } ) )
59	58	oveq1d 6055	. . 3 |- ( ph -> ( ( N ` { ( X .- ( ( inv g ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) = ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) )
60	55, 59	ineq12d 3503	. 2 |- ( ph -> ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( inv g ` W ) ` Z ) } ) ) i^i ( ( N ` { ( X .- ( ( inv g ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) ) = ( ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) i^i ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) ) )
61	13, 54, 60	3eqtrd 2440	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 359   = wceq 1649   e. wcel 1721   =/= wne 2567   \ cdif 3277   i^i cin 3279   {csn 3774   {cpr 3775   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641   -gcsg 14643   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   LVecclvec 16129

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130