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

Theorem baerlem5bmN 36915
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 36916 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 3493 . . . . 5  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . 6  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3493 . . . . 5  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . 6  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . 6  |-  .+  =  ( +g  `  W )
7 eqid 2467 . . . . . 6  |-  ( invg `  W )  =  ( invg `  W )
8 baerlem3.m . . . . . 6  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 15965 . . . . 5  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
102, 4, 9syl2anc 661 . . . 4  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
1110sneqd 4045 . . 3  |-  ( ph  ->  { ( Y  .-  Z ) }  =  { ( Y  .+  ( ( invg `  W ) `  Z
) ) } )
1211fveq2d 5876 . 2  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( N `  {
( Y  .+  (
( invg `  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 17623 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
1916, 18syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
205, 7lmodvnegcl 17422 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( invg `  W ) `  Z
)  e.  V )
2119, 4, 20syl2anc 661 . . . 4  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  V )
22 eqid 2467 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
235, 22, 15, 19, 2, 4lspprcl 17495 . . . . . 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 17469 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
265, 15, 16, 17, 2, 4, 24lspindpi 17649 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2726simpld 459 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
285, 13, 15, 16, 25, 2, 27lspsnne1 17634 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
29 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3029necomd 2738 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
315, 13, 15, 16, 3, 2, 30lspsnne1 17634 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
325, 15, 16, 17, 4, 2, 31, 24lspexchn2 17648 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
33 lmodgrp 17390 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3419, 33syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3534adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
364adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
375, 7grpinvinv 15977 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
3835, 36, 37syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
3919adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
405, 22, 15, 19, 2, 17lspprcl 17495 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4140adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
42 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4322, 7lssvnegcl 17473 . . . . . . 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 } ) )
4439, 41, 42, 43syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4538, 44eqeltrrd 2556 . . . . 5  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4632, 45mtand 659 . . . 4  |-  ( ph  ->  -.  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
475, 15, 16, 21, 17, 2, 28, 46lspexchn2 17648 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( invg `  W ) `
 Z ) } ) )
485, 7, 15lspsnneg 17523 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
4919, 4, 48syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  {
( ( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5029, 49neeqtrrd 2767 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( invg `  W ) `  Z
) } ) )
515, 13, 7grpinvnzcl 15982 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( invg `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5234, 3, 51syl2anc 661 . . 3  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
535, 8, 13, 14, 15, 16, 17, 47, 50, 1, 52, 6baerlem5b 36913 . 2  |-  ( ph  ->  ( N `  {
( Y  .+  (
( invg `  W ) `  Z
) ) } )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { ( ( invg `  W
) `  Z ) } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) ) ) )
5449oveq2d 6311 . . 3  |-  ( ph  ->  ( ( N `  { Y } )  .(+)  ( N `  { ( ( invg `  W ) `  Z
) } ) )  =  ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) ) )
5510eqcomd 2475 . . . . . . 7  |-  ( ph  ->  ( Y  .+  (
( invg `  W ) `  Z
) )  =  ( Y  .-  Z ) )
5655oveq2d 6311 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) )  =  ( X 
.-  ( Y  .-  Z ) ) )
5756sneqd 4045 . . . . 5  |-  ( ph  ->  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) }  =  { ( X  .-  ( Y  .-  Z ) ) } )
5857fveq2d 5876 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )  =  ( N `  {
( X  .-  ( Y  .-  Z ) ) } ) )
5958oveq1d 6310 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) )  =  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) )
6054, 59ineq12d 3706 . 2  |-  ( ph  ->  ( ( ( N `
 { Y }
)  .(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )  .(+)  ( N `  { X } ) ) )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .-  Z ) ) } )  .(+)  ( N `
 { X }
) ) ) )
6112, 53, 603eqtrd 2512 1  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    i^i cin 3480   {csn 4033   {cpr 4035   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926   -gcsg 15927   LSSumclsm 16527   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488   LVecclvec 17619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator