Theorem baerlem5amN 35355

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 35357 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 3402	. . . . . 6 |- ( ph -> Y e. V )
3		baerlem3.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
4	3	eldifad 3402	. . . . . 6 |- ( ph -> Z e. V )
5		baerlem3.v	. . . . . . 7 |- V = ( Base ` W )
6		baerlem5a.p	. . . . . . 7 |- .+ = ( +g ` W )
7		eqid 2471	. . . . . . 7 |- ( invg ` W ) = ( invg ` W )
8		baerlem3.m	. . . . . . 7 |- .- = ( -g ` W )
9	5, 6, 7, 8	grpsubval 16787	. . . . . 6 |- ( ( Y e. V /\ Z e. V ) -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
10	2, 4, 9	syl2anc 673	. . . . 5 |- ( ph -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
11	10	oveq2d 6324	. . . 4 |- ( ph -> ( X .- ( Y .- Z ) ) = ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) )
12	11	sneqd 3971	. . 3 |- ( ph -> { ( X .- ( Y .- Z ) ) } = { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } )
13	12	fveq2d 5883	. 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 18407	. . . . . 6 |- ( W e. LVec -> W e. LMod )
20	17, 19	syl 17	. . . . 5 |- ( ph -> W e. LMod )
21	5, 7	lmodvnegcl 18207	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( ( invg ` W ) ` Z ) e. V )
22	20, 4, 21	syl2anc 673	. . . 4 |- ( ph -> ( ( invg ` W ) ` Z ) e. V )
23		eqid 2471	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
24	5, 23, 16, 20, 2, 4	lspprcl 18279	. . . . . 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 18253	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
27	5, 16, 17, 18, 2, 4, 25	lspindpi 18433	. . . . . 6 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
28	27	simpld 466	. . . . 5 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
29	5, 14, 16, 17, 26, 2, 28	lspsnne1 18418	. . . 4 |- ( ph -> -. X e. ( N ` { Y } ) )
30		baerlem3.d	. . . . . . . 8 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
31	30	necomd 2698	. . . . . . 7 |- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )
32	5, 14, 16, 17, 3, 2, 31	lspsnne1 18418	. . . . . 6 |- ( ph -> -. Z e. ( N ` { Y } ) )
33	5, 16, 17, 18, 4, 2, 32, 25	lspexchn2 18432	. . . . 5 |- ( ph -> -. Z e. ( N ` { Y , X } ) )
34		lmodgrp 18176	. . . . . . . . 9 |- ( W e. LMod -> W e. Grp )
35	17, 19, 34	3syl 18	. . . . . . . 8 |- ( ph -> W e. Grp )
36	35	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. Grp )
37	4	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. V )
38	5, 7	grpinvinv 16799	. . . . . . 7 |- ( ( W e. Grp /\ Z e. V ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
39	36, 37, 38	syl2anc 673	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
40	20	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. LMod )
41	5, 23, 16, 20, 2, 18	lspprcl 18279	. . . . . . . 8 |- ( ph -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
42	41	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
43		simpr 468	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
44	23, 7	lssvnegcl 18257	. . . . . . 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 1292	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
46	39, 45	eqeltrrd 2550	. . . . 5 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. ( N ` { Y , X } ) )
47	33, 46	mtand 671	. . . 4 |- ( ph -> -. ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
48	5, 16, 17, 22, 18, 2, 29, 47	lspexchn2 18432	. . 3 |- ( ph -> -. X e. ( N ` { Y , ( ( invg ` W ) ` Z ) } ) )
49	5, 7, 16	lspsnneg 18307	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
50	20, 4, 49	syl2anc 673	. . . 4 |- ( ph -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
51	30, 50	neeqtrrd 2717	. . 3 |- ( ph -> ( N ` { Y } ) =/= ( N ` { ( ( invg ` W ) ` Z ) } ) )
52	5, 14, 7	grpinvnzcl 16804	. . . 4 |- ( ( W e. Grp /\ Z e. ( V \ { .0. } ) ) -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
53	35, 3, 52	syl2anc 673	. . 3 |- ( ph -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
54	5, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6	baerlem5a 35353	. 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 6324	. . 3 |- ( ph -> ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) = ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) )
56	5, 6, 8, 7, 35, 18, 4	grpsubinv 16805	. . . . . 6 |- ( ph -> ( X .- ( ( invg ` W ) ` Z ) ) = ( X .+ Z ) )
57	56	sneqd 3971	. . . . 5 |- ( ph -> { ( X .- ( ( invg ` W ) ` Z ) ) } = { ( X .+ Z ) } )
58	57	fveq2d 5883	. . . 4 |- ( ph -> ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) = ( N ` { ( X .+ Z ) } ) )
59	58	oveq1d 6323	. . 3 |- ( ph -> ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) = ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) )
60	55, 59	ineq12d 3626	. 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 2509	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 376   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   i^i cin 3389   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   0gc0g 15416   Grpcgrp 16747   invgcminusg 16748   -gcsg 16749   LSSumclsm 17364   LModclmod 18169   LSubSpclss 18233   LSpanclspn 18272   LVecclvec 18403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-lsm 17366  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404

This theorem is referenced by: (None)