Theorem baerlem5amN 37586

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 37588 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 3483	. . . . . 6 |- ( ph -> Y e. V )
3		baerlem3.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
4	3	eldifad 3483	. . . . . 6 |- ( ph -> Z e. V )
5		baerlem3.v	. . . . . . 7 |- V = ( Base ` W )
6		baerlem5a.p	. . . . . . 7 |- .+ = ( +g ` W )
7		eqid 2457	. . . . . . 7 |- ( invg ` W ) = ( invg ` W )
8		baerlem3.m	. . . . . . 7 |- .- = ( -g ` W )
9	5, 6, 7, 8	grpsubval 16220	. . . . . 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 6312	. . . 4 |- ( ph -> ( X .- ( Y .- Z ) ) = ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) )
12	11	sneqd 4044	. . 3 |- ( ph -> { ( X .- ( Y .- Z ) ) } = { ( X .- ( Y .+ ( ( invg ` W ) ` Z ) ) ) } )
13	12	fveq2d 5876	. 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 17879	. . . . . 6 |- ( W e. LVec -> W e. LMod )
20	17, 19	syl 16	. . . . 5 |- ( ph -> W e. LMod )
21	5, 7	lmodvnegcl 17678	. . . . 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 2457	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
24	5, 23, 16, 20, 2, 4	lspprcl 17751	. . . . . 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 17725	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
27	5, 16, 17, 18, 2, 4, 25	lspindpi 17905	. . . . . 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 17890	. . . 4 |- ( ph -> -. X e. ( N ` { Y } ) )
30		baerlem3.d	. . . . . . . 8 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
31	30	necomd 2728	. . . . . . 7 |- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )
32	5, 14, 16, 17, 3, 2, 31	lspsnne1 17890	. . . . . 6 |- ( ph -> -. Z e. ( N ` { Y } ) )
33	5, 16, 17, 18, 4, 2, 32, 25	lspexchn2 17904	. . . . 5 |- ( ph -> -. Z e. ( N ` { Y , X } ) )
34		lmodgrp 17646	. . . . . . . . 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 16232	. . . . . . 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 17751	. . . . . . . 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 17729	. . . . . . 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 1228	. . . . . 6 |- ( ( ph /\ ( ( invg ` W ) ` Z ) e. ( N ` { Y , X } ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` Z ) ) e. ( N ` { Y , X } ) )
46	39, 45	eqeltrrd 2546	. . . . 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 17904	. . 3 |- ( ph -> -. X e. ( N ` { Y , ( ( invg ` W ) ` Z ) } ) )
49	5, 7, 16	lspsnneg 17779	. . . . 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 2757	. . 3 |- ( ph -> ( N ` { Y } ) =/= ( N ` { ( ( invg ` W ) ` Z ) } ) )
52	5, 14, 7	grpinvnzcl 16237	. . . 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 37584	. 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 6312	. . 3 |- ( ph -> ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { ( ( invg ` W ) ` Z ) } ) ) = ( ( N ` { ( X .- Y ) } ) .(+) ( N ` { Z } ) ) )
56	5, 6, 8, 7, 35, 18, 4	grpsubinv 16238	. . . . . 6 |- ( ph -> ( X .- ( ( invg ` W ) ` Z ) ) = ( X .+ Z ) )
57	56	sneqd 4044	. . . . 5 |- ( ph -> { ( X .- ( ( invg ` W ) ` Z ) ) } = { ( X .+ Z ) } )
58	57	fveq2d 5876	. . . 4 |- ( ph -> ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) = ( N ` { ( X .+ Z ) } ) )
59	58	oveq1d 6311	. . 3 |- ( ph -> ( ( N ` { ( X .- ( ( invg ` W ) ` Z ) ) } ) .(+) ( N ` { Y } ) ) = ( ( N ` { ( X .+ Z ) } ) .(+) ( N ` { Y } ) ) )
60	55, 59	ineq12d 3697	. 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 2502	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 1395   e. wcel 1819   =/= wne 2652   \ cdif 3468   i^i cin 3470   {csn 4032   {cpr 4034   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   0gc0g 14857   Grpcgrp 16180   invgcminusg 16181   -gcsg 16182   LSSumclsm 16781   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   LVecclvec 17875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-cntz 16482  df-lsm 16783  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876

This theorem is referenced by: (None)