Theorem baerlem5amN 37005

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 37007 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 3586	. . . . . 6 |- ( ph -> Y e. V )
3		baerlem3.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
4	3	eldifad 3586	. . . . . 6 |- ( ph -> Z e. V )
5		baerlem3.v	. . . . . . 7 |- V = ( Base ` W )
6		baerlem5a.p	. . . . . . 7 |- .+ = ( +g ` W )
7		eqid 2622	. . . . . . 7 |- ( invg ` W ) = ( invg ` W )
8		baerlem3.m	. . . . . . 7 |- .- = ( -g ` W )
9	5, 6, 7, 8	grpsubval 17465	. . . . . 6 |- ( ( Y e. V /\ Z e. V ) -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
10	2, 4, 9	syl2anc 693	. . . . 5 |- ( ph -> ( Y .- Z ) = ( Y .+ ( ( invg ` W ) ` Z ) ) )
11	10	oveq2d 6666	. . . 4 |- ( ph -> ( X .- ( Y .- Z ) ) = ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) )
12	11	sneqd 4189	. . 3 |- ( ph -> { ( X .- ( Y .- Z ) ) } = { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } )
13	12	fveq2d 6195	. 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 19106	. . . . . 6 |- ( W e. LVec -> W e. LMod )
20	17, 19	syl 17	. . . . 5 |- ( ph -> W e. LMod )
21	5, 7	lmodvnegcl 18904	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( ( invg ` W ) ` Z ) e. V )
22	20, 4, 21	syl2anc 693	. . . 4 |- ( ph -> ( ( invg ` W ) ` Z ) e. V )
23		eqid 2622	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
24	5, 23, 16, 20, 2, 4	lspprcl 18978	. . . . . 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 18952	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
27	5, 16, 17, 18, 2, 4, 25	lspindpi 19132	. . . . . 6 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
28	27	simpld 475	. . . . 5 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
29	5, 14, 16, 17, 26, 2, 28	lspsnne1 19117	. . . 4 |- ( ph -> -. X e. ( N ` { Y } ) )
30		baerlem3.d	. . . . . . . 8 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
31	30	necomd 2849	. . . . . . 7 |- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )
32	5, 14, 16, 17, 3, 2, 31	lspsnne1 19117	. . . . . 6 |- ( ph -> -. Z e. ( N ` { Y } ) )
33	5, 16, 17, 18, 4, 2, 32, 25	lspexchn2 19131	. . . . 5 |- ( ph -> -. Z e. ( N ` { Y , X } ) )
34		lmodgrp 18870	. . . . . . . . 9 |- ( W e. LMod -> W e. Grp )
35	17, 19, 34	3syl 18	. . . . . . . 8 |- ( ph -> W e. Grp )
36	35	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. Grp )
37	4	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. V )
38	5, 7	grpinvinv 17482	. . . . . . 7 |- ( ( W e. Grp /\ Z e. V ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
39	36, 37, 38	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) = Z )
40	20	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> W e. LMod )
41	5, 23, 16, 20, 2, 18	lspprcl 18978	. . . . . . . 8 |- ( ph -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
42	41	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( N ` { Y , X } ) e. ( LSubSp ` W ) )
43		simpr 477	. . . . . . 7 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
44	23, 7	lssvnegcl 18956	. . . . . . 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 1326	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
46	39, 45	eqeltrrd 2702	. . . . 5 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> Z e. ( N ` { Y , X } ) )
47	33, 46	mtand 691	. . . 4 |- ( ph -> -. ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) )
48	5, 16, 17, 22, 18, 2, 29, 47	lspexchn2 19131	. . 3 |- ( ph -> -. X e. ( N ` { Y , ( ( invg ` W ) ` Z ) } ) )
49	5, 7, 16	lspsnneg 19006	. . . . 5 |- ( ( W e. LMod /\ Z e. V ) -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
50	20, 4, 49	syl2anc 693	. . . 4 |- ( ph -> ( N ` { ( ( invg ` W ) ` Z ) } ) = ( N ` { Z } ) )
51	30, 50	neeqtrrd 2868	. . 3 |- ( ph -> ( N ` { Y } ) =/= ( N ` { ( ( invg ` W ) ` Z ) } ) )
52	5, 14, 7	grpinvnzcl 17487	. . . 4 |- ( ( W e. Grp /\ Z e. ( V \ { .0. } ) ) -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
53	35, 3, 52	syl2anc 693	. . 3 |- ( ph -> ( ( invg ` W ) ` Z ) e. ( V \ { .0. } ) )
54	5, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6	baerlem5a 37003	. 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 6666	. . 3 |- ( ph -> ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) = ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) )
56	5, 6, 8, 7, 35, 18, 4	grpsubinv 17488	. . . . . 6 |- ( ph -> ( X .- ( ( invg ` W ) ` Z ) ) = ( X .+ Z ) )
57	56	sneqd 4189	. . . . 5 |- ( ph -> { ( X .- ( ( invg ` W ) ` Z ) ) } = { ( X .+ Z ) } )
58	57	fveq2d 6195	. . . 4 |- ( ph -> ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) = ( N ` { ( X .+ Z ) } ) )
59	58	oveq1d 6665	. . 3 |- ( ph -> ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) = ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) )
60	55, 59	ineq12d 3815	. 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 2660	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 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103

This theorem is referenced by: (None)