Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  baerlem5bmN Unicode version

Theorem baerlem5bmN 32354
Description: An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 32355 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
baerlem5bmN  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )

Proof of Theorem baerlem5bmN
StepHypRef Expression
1 baerlem3.y . . . . . 6  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
21eldifad 3324 . . . . 5  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . 6  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3324 . . . . 5  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . 6  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . 6  |-  .+  =  ( +g  `  W )
7 eqid 2435 . . . . . 6  |-  ( inv g `  W )  =  ( inv g `  W )
8 baerlem3.m . . . . . 6  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 14836 . . . . 5  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
102, 4, 9syl2anc 643 . . . 4  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1110sneqd 3819 . . 3  |-  ( ph  ->  { ( Y  .-  Z ) }  =  { ( Y  .+  ( ( inv g `  W ) `  Z
) ) } )
1211fveq2d 5723 . 2  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( N `  {
( Y  .+  (
( inv g `  W ) `  Z
) ) } ) )
13 baerlem3.o . . 3  |-  .0.  =  ( 0g `  W )
14 baerlem3.s . . 3  |-  .(+)  =  (
LSSum `  W )
15 baerlem3.n . . 3  |-  N  =  ( LSpan `  W )
16 baerlem3.w . . 3  |-  ( ph  ->  W  e.  LVec )
17 baerlem3.x . . 3  |-  ( ph  ->  X  e.  V )
18 lveclmod 16166 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
1916, 18syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
205, 7lmodvnegcl 15973 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2119, 4, 20syl2anc 643 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
22 eqid 2435 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
235, 22, 15, 19, 2, 4lspprcl 16042 . . . . . 6  |-  ( ph  ->  ( N `  { Y ,  Z }
)  e.  ( LSubSp `  W ) )
24 baerlem3.c . . . . . 6  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
255, 13, 22, 19, 23, 17, 24lssneln0 16016 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
265, 15, 16, 17, 2, 4, 24lspindpi 16192 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2726simpld 446 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
285, 13, 15, 16, 25, 2, 27lspsnne1 16177 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
29 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3029necomd 2681 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
315, 13, 15, 16, 3, 2, 30lspsnne1 16177 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
325, 15, 16, 17, 4, 2, 31, 24lspexchn2 16191 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
33 lmodgrp 15945 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3419, 33syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3534adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
364adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
375, 7grpinvinv 14846 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
3835, 36, 37syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
3919adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
405, 22, 15, 19, 2, 17lspprcl 16042 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4140adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
42 simpr 448 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4322, 7lssvnegcl 16020 . . . . . . 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 } ) )
4439, 41, 42, 43syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4538, 44eqeltrrd 2510 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4632, 45mtand 641 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
475, 15, 16, 21, 17, 2, 28, 46lspexchn2 16191 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
485, 7, 15lspsnneg 16070 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
4919, 4, 48syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5029, 49neeqtrrd 2622 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
515, 13, 7grpinvnzcl 14851 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5234, 3, 51syl2anc 643 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
535, 8, 13, 14, 15, 16, 17, 47, 50, 1, 52, 6baerlem5b 32352 . 2  |-  ( ph  ->  ( N `  {
( Y  .+  (
( inv g `  W ) `  Z
) ) } )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { ( ( inv g `  W
) `  Z ) } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) ) ) )
5449oveq2d 6088 . . 3  |-  ( ph  ->  ( ( N `  { Y } )  .(+)  ( N `  { ( ( inv g `  W ) `  Z
) } ) )  =  ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) ) )
5510eqcomd 2440 . . . . . . 7  |-  ( ph  ->  ( Y  .+  (
( inv g `  W ) `  Z
) )  =  ( Y  .-  Z ) )
5655oveq2d 6088 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) )  =  ( X 
.-  ( Y  .-  Z ) ) )
5756sneqd 3819 . . . . 5  |-  ( ph  ->  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) }  =  { ( X  .-  ( Y  .-  Z ) ) } )
5857fveq2d 5723 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  =  ( N `  {
( X  .-  ( Y  .-  Z ) ) } ) )
5958oveq1d 6087 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) )  =  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) )
6054, 59ineq12d 3535 . 2  |-  ( ph  ->  ( ( ( N `
 { Y }
)  .(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  .(+)  ( N `  { X } ) ) )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .-  Z ) ) } )  .(+)  ( N `
 { X }
) ) ) )
6112, 53, 603eqtrd 2471 1  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    i^i cin 3311   {csn 3806   {cpr 3807   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517   0gc0g 13711   Grpcgrp 14673   inv gcminusg 14674   -gcsg 14676   LSSumclsm 15256   LModclmod 15938   LSubSpclss 15996   LSpanclspn 16035   LVecclvec 16162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-tpos 6470  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-0g 13715  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163
  Copyright terms: Public domain W3C validator